(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C. *) From HB Require Import structures. From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval finmap. Require Import mathcomp_extra boolp classical_sets signed functions cardinality. Require Import reals ereal topology normedtype sequences esum measure. Require Import lebesgue_measure fsbigop numfun. (******************************************************************************) (* Lebesgue Integral *) (* *) (* This file contains a formalization of the Lebesgue integral. It starts *) (* with simple functions and their integral, provides basic operations *) (* (addition, etc.), and proves the properties of their integral *) (* (semi-linearity, non-decreasingness). It then defines the integral of *) (* measurable functions, proves the approximation theorem, the properties of *) (* their integral (semi-linearity, non-decreasingness), the monotone *) (* convergence theorem, and Fatou's lemma. Finally, it proves the linearity *) (* properties of the integral, the dominated convergence theorem and Fubini's *) (* theorem. *) (* *) (* Main reference: *) (* - Daniel Li, Intégration et applications, 2016 *) (* *) (* {nnfun T >-> R} == type of non-negative functions *) (* {fimfun T >-> R} == type of functions with a finite image *) (* {sfun T >-> R} == type of simple functions *) (* {nnsfun T >-> R} == type of non-negative simple functions *) (* cst_nnsfun r == constant simple function *) (* nnsfun0 := cst_nnsfun 0 *) (* sintegral mu f == integral of the function f with the measure mu *) (* \int[mu]_(x in D) f x == integral of the measurable function f over the *) (* domain D with measure mu *) (* \int[mu]_x f x := \int[mu]_(x in setT) f x *) (* dyadic_itv n k == the interval *) (* `[(k%:R * 2 ^- n), (k.+1%:R * 2 ^- n)[ *) (* approx D f == nondecreasing sequence of functions that *) (* approximates f over D using dyadic intervals *) (* Rintegral mu D f := fine (\int[mu]_(x in D) f x). *) (* mu.-integrable D f == f is measurable over D and the integral of f *) (* w.r.t. D is < +oo *) (* ae_eq D f g == f is equal to g almost everywhere *) (* product_measure1 m1 s2 == product measure over T1 * T2, m1 is a measure *) (* measure over T1, s2 is a proof that a measure m2 *) (* over T2 is sigma-finite *) (* product_measure2 s2 m2 == product_measure1 mutatis mutandis *) (* *) (******************************************************************************) Set Implicit Arguments. Unset Strict Implicit. Unset Printing Implicit Defensive. Import Order.TTheory GRing.Theory Num.Def Num.Theory. Import numFieldTopology.Exports. Local Open Scope classical_set_scope. Local Open Scope ring_scope. Reserved Notation "\int [ mu ]_ ( i 'in' D ) F" (at level 36, F at level 36, mu at level 10, i, D at level 50, format "'[' \int [ mu ]_ ( i 'in' D ) '/ ' F ']'"). Reserved Notation "\int [ mu ]_ i F" (at level 36, F at level 36, mu at level 10, i at level 0, right associativity, format "'[' \int [ mu ]_ i '/ ' F ']'"). Reserved Notation "mu .-integrable" (at level 2, format "mu .-integrable"). #[global] Hint Extern 0 (measurable [set _]) => solve [apply: measurable_set1] : core. HB.mixin Record IsMeasurableFun d (aT : measurableType d) (rT : realType) (f : aT -> rT) := { measurable_funP : measurable_fun setT f }. #[global] Hint Resolve fimfun_inP : core. HB.structure Definition MeasurableFun d aT rT := {f of @IsMeasurableFun d aT rT f}. Reserved Notation "{ 'mfun' aT >-> T }" (at level 0, format "{ 'mfun' aT >-> T }"). Reserved Notation "[ 'mfun' 'of' f ]" (at level 0, format "[ 'mfun' 'of' f ]"). Notation "{ 'mfun' aT >-> T }" := (@MeasurableFun.type _ aT T) : form_scope. Notation "[ 'mfun' 'of' f ]" := [the {mfun _ >-> _} of f] : form_scope. #[global] Hint Resolve measurable_funP : core. HB.structure Definition SimpleFun d (aT (*rT*) : measurableType d) (rT : realType) := {f of @IsMeasurableFun d aT rT f & @FiniteImage aT rT f}. Reserved Notation "{ 'sfun' aT >-> T }" (at level 0, format "{ 'sfun' aT >-> T }"). Reserved Notation "[ 'sfun' 'of' f ]" (at level 0, format "[ 'sfun' 'of' f ]"). Notation "{ 'sfun' aT >-> T }" := (@SimpleFun.type _ aT T) : form_scope. Notation "[ 'sfun' 'of' f ]" := [the {sfun _ >-> _} of f] : form_scope. Lemma measurable_sfunP {d} {aT : measurableType d} {rT : realType} (f : {mfun aT >-> rT}) (y : rT) : measurable (f @^-1` [set y]). Proof. by rewrite -[f @^-1` _]setTI; exact: measurable_funP. Qed. HB.mixin Record IsNonNegFun (aT : Type) (rT : numDomainType) (f : aT -> rT) := { fun_ge0 : forall x, 0 <= f x }. HB.structure Definition NonNegFun aT rT := {f of @IsNonNegFun aT rT f}. Reserved Notation "{ 'nnfun' aT >-> T }" (at level 0, format "{ 'nnfun' aT >-> T }"). Reserved Notation "[ 'nnfun' 'of' f ]" (at level 0, format "[ 'nnfun' 'of' f ]"). Notation "{ 'nnfun' aT >-> T }" := (@NonNegFun.type aT T) : form_scope. Notation "[ 'nnfun' 'of' f ]" := [the {nnfun _ >-> _} of f] : form_scope. #[global] Hint Extern 0 (is_true (0 <= _)) => solve [apply: fun_ge0] : core. HB.structure Definition NonNegSimpleFun d (aT : measurableType d) (rT : realType) := {f of @SimpleFun d _ _ f & @NonNegFun aT rT f}. Reserved Notation "{ 'nnsfun' aT >-> T }" (at level 0, format "{ 'nnsfun' aT >-> T }"). Reserved Notation "[ 'nnsfun' 'of' f ]" (at level 0, format "[ 'nnsfun' 'of' f ]"). Notation "{ 'nnsfun' aT >-> T }" := (@NonNegSimpleFun.type _ aT T) : form_scope. Notation "[ 'nnsfun' 'of' f ]" := [the {nnsfun _ >-> _} of f] : form_scope. Section ring. Context (aT : pointedType) (rT : ringType). Lemma fimfun_mulr_closed : mulr_closed (@fimfun aT rT). Proof. split=> [|f g]; rewrite !inE/=; first exact: finite_image_cst. by move=> fA gA; apply: (finite_image11 (fun x y => x * y)). Qed. Canonical fimfun_mul := MulrPred fimfun_mulr_closed. Canonical fimfun_ring := SubringPred fimfun_mulr_closed. Definition fimfun_ringMixin := [ringMixin of {fimfun aT >-> rT} by <:]. Canonical fimfun_ringType := RingType {fimfun aT >-> rT} fimfun_ringMixin. Implicit Types (f g : {fimfun aT >-> rT}). Lemma fimfunM f g : f * g = f \* g :> (_ -> _). Proof. by []. Qed. Lemma fimfun1 : (1 : {fimfun aT >-> rT}) = cst 1 :> (_ -> _). Proof. by []. Qed. Lemma fimfun_prod I r (P : {pred I}) (f : I -> {fimfun aT >-> rT}) (x : aT) : (\sum_(i <- r | P i) f i) x = \sum_(i <- r | P i) f i x. Proof. by elim/big_rec2: _ => //= i y ? Pi <-. Qed. Lemma fimfunX f n : f ^+ n = (fun x => f x ^+ n) :> (_ -> _). Proof. by apply/funext => x; elim: n => [|n IHn]//; rewrite !exprS fimfunM/= IHn. Qed. Lemma indic_fimfun_subproof X : @FiniteImage aT rT \1_X. Proof. split; apply: (finite_subfset [fset 0; 1]%fset) => x [tt /=]. by rewrite !inE indicE; case: (_ \in _) => _ <-; rewrite ?eqxx ?orbT. Qed. HB.instance Definition _ X := indic_fimfun_subproof X. Definition indic_fimfun (X : set aT) := [the {fimfun aT >-> rT} of \1_X]. HB.instance Definition _ k f := FImFun.copy (k \o* f) (f * cst_fimfun k). Definition scale_fimfun k f := [the {fimfun aT >-> rT} of k \o* f]. End ring. Arguments indic_fimfun {aT rT} _. Section comring. Context (aT : pointedType) (rT : comRingType). Definition fimfun_comRingMixin := [comRingMixin of {fimfun aT >-> rT} by <:]. Canonical fimfun_comRingType := ComRingType {fimfun aT >-> rT} fimfun_comRingMixin. Implicit Types (f g : {fimfun aT >-> rT}). HB.instance Definition _ f g := FImFun.copy (f \* g) (f * g). End comring. Lemma fimfunE T (R : ringType) (f : {fimfun T >-> R}) x : f x = \sum_(y <- fset_set (range f)) (y * \1_(f @^-1` [set y]) x). Proof. have fxfA: f x \in fset_set (f @` setT) by rewrite in_fset_set// inE; exists x. rewrite (big_fsetD1 (f x))//= indicE (@id (_ \in _)) ?mulr1 ?inE//=. rewrite big_seq_cond ?big1 ?addr0// => y; rewrite ?andbT !inE eq_sym. move=> /andP[fxNy yA]; rewrite indicE [_ \in _]negbTE ?mulr0// notin_set. by move=> fxy; rewrite -fxy eqxx in fxNy. Qed. Lemma fimfunEord T (R : ringType) (f : {fimfun T >-> R}) (s := fset_set (f @` setT)) : forall x, f x = \sum_(i < #|`s|) (s`_i * \1_(f @^-1` [set s`_i]) x). Proof. by move=> x; rewrite fimfunE /s // (big_nth 0) big_mkord. Qed. Lemma trivIset_preimage1 {aT rT} D (f : aT -> rT) : trivIset D (fun x => f @^-1` [set x]). Proof. by move=> y z _ _ [x [<- <-]]. Qed. Lemma trivIset_preimage1_in {aT} {rT : choiceType} (D : set rT) (A : set aT) (f : aT -> rT) : trivIset D (fun x => A `&` f @^-1` [set x]). Proof. by move=> y z _ _ [x [[_ <-] [_ <-]]]. Qed. Section fimfun_bin. Variables (d : measure_display) (T : measurableType d). Variables (R : numDomainType) (f g : {fimfun T >-> R}). Lemma max_fimfun_subproof : @FiniteImage T R (f \max g). Proof. by split; apply: (finite_image11 maxr). Qed. HB.instance Definition _ := max_fimfun_subproof. End fimfun_bin. HB.factory Record FiniteDecomp (T : pointedType) (R : ringType) (f : T -> R) := { fimfunE : exists (r : seq R) (A_ : R -> set T), forall x, f x = \sum_(y <- r) (y * \1_(A_ y) x) }. HB.builders Context T R f of @FiniteDecomp T R f. Lemma finite_subproof: @FiniteImage T R f. Proof. split; have [r [A_ fE]] := fimfunE. suff -> : f = \sum_(y <- r) cst_fimfun y * indic_fimfun (A_ y) by []. by apply/funext=> x; rewrite fE fimfun_sum. Qed. HB.instance Definition _ := finite_subproof. HB.end. Section mfun_pred. Context {d} {aT : measurableType d} {rT : realType}. Definition mfun : {pred aT -> rT} := mem [set f | measurable_fun setT f]. Definition mfun_key : pred_key mfun. Proof. exact. Qed. Canonical mfun_keyed := KeyedPred mfun_key. End mfun_pred. Section mfun. Context {d} {aT : measurableType d} {rT : realType}. Notation T := {mfun aT >-> rT}. Notation mfun := (@mfun _ aT rT). Section Sub. Context (f : aT -> rT) (fP : f \in mfun). Definition mfun_Sub_subproof := @IsMeasurableFun.Build d aT rT f (set_mem fP). #[local] HB.instance Definition _ := mfun_Sub_subproof. Definition mfun_Sub := [mfun of f]. End Sub. Lemma mfun_rect (K : T -> Type) : (forall f (Pf : f \in mfun), K (mfun_Sub Pf)) -> forall u : T, K u. Proof. move=> Ksub [f [[Pf]]]/=. by suff -> : Pf = (set_mem (@mem_set _ [set f | _] f Pf)) by apply: Ksub. Qed. Lemma mfun_valP f (Pf : f \in mfun) : mfun_Sub Pf = f :> (_ -> _). Proof. by []. Qed. Canonical mfun_subType := SubType T _ _ mfun_rect mfun_valP. Lemma mfuneqP (f g : {mfun aT >-> rT}) : f = g <-> f =1 g. Proof. by split=> [->//|fg]; apply/val_inj/funext. Qed. Definition mfuneqMixin := [eqMixin of {mfun aT >-> rT} by <:]. Canonical mfuneqType := EqType {mfun aT >-> rT} mfuneqMixin. Definition mfunchoiceMixin := [choiceMixin of {mfun aT >-> rT} by <:]. Canonical mfunchoiceType := ChoiceType {mfun aT >-> rT} mfunchoiceMixin. Lemma cst_mfun_subproof x : @IsMeasurableFun d aT rT (cst x). Proof. by split; apply: measurable_fun_cst. Qed. HB.instance Definition _ x := @cst_mfun_subproof x. Definition cst_mfun x := [the {mfun aT >-> rT} of cst x]. Lemma mfun_cst x : @cst_mfun x =1 cst x. Proof. by []. Qed. End mfun. Section ring. Context (d : measure_display) (aT : measurableType d) (rT : realType). Lemma mfun_subring_closed : subring_closed (@mfun _ aT rT). Proof. split=> [|f g|f g]; rewrite !inE/=. - exact: measurable_fun_cst. - exact: measurable_funB. - exact: measurable_funM. Qed. Canonical mfun_add := AddrPred mfun_subring_closed. Canonical mfun_zmod := ZmodPred mfun_subring_closed. Canonical mfun_mul := MulrPred mfun_subring_closed. Canonical mfun_subring := SubringPred mfun_subring_closed. Definition mfun_zmodMixin := [zmodMixin of {mfun aT >-> rT} by <:]. Canonical mfun_zmodType := ZmodType {mfun aT >-> rT} mfun_zmodMixin. Definition mfun_ringMixin := [ringMixin of {mfun aT >-> rT} by <:]. Canonical mfun_ringType := RingType {mfun aT >-> rT} mfun_ringMixin. Definition mfun_comRingMixin := [comRingMixin of {mfun aT >-> rT} by <:]. Canonical mfun_comRingType := ComRingType {mfun aT >-> rT} mfun_comRingMixin. Implicit Types (f g : {mfun aT >-> rT}). Lemma mfun0 : (0 : {mfun aT >-> rT}) =1 cst 0 :> (_ -> _). Proof. by []. Qed. Lemma mfun1 : (1 : {mfun aT >-> rT}) =1 cst 1 :> (_ -> _). Proof. by []. Qed. Lemma mfunN f : - f = \- f :> (_ -> _). Proof. by []. Qed. Lemma mfunD f g : f + g = f \+ g :> (_ -> _). Proof. by []. Qed. Lemma mfunB f g : f - g = f \- g :> (_ -> _). Proof. by []. Qed. Lemma mfunM f g : f * g = f \* g :> (_ -> _). Proof. by []. Qed. Lemma mfun_sum I r (P : {pred I}) (f : I -> {mfun aT >-> rT}) (x : aT) : (\sum_(i <- r | P i) f i) x = \sum_(i <- r | P i) f i x. Proof. by elim/big_rec2: _ => //= i y ? Pi <-. Qed. Lemma mfun_prod I r (P : {pred I}) (f : I -> {mfun aT >-> rT}) (x : aT) : (\sum_(i <- r | P i) f i) x = \sum_(i <- r | P i) f i x. Proof. by elim/big_rec2: _ => //= i y ? Pi <-. Qed. Lemma mfunX f n : f ^+ n = (fun x => f x ^+ n) :> (_ -> _). Proof. by apply/funext=> x; elim: n => [|n IHn]//; rewrite !exprS mfunM/= IHn. Qed. HB.instance Definition _ f g := MeasurableFun.copy (f \+ g) (f + g). HB.instance Definition _ f g := MeasurableFun.copy (\- f) (- f). HB.instance Definition _ f g := MeasurableFun.copy (f \- g) (f - g). HB.instance Definition _ f g := MeasurableFun.copy (f \* g) (f * g). Definition mindic (D : set aT) of measurable D : aT -> rT := \1_D. Lemma mindicE (D : set aT) (mD : measurable D) : mindic mD = (fun x => (x \in D)%:R). Proof. by rewrite /mindic funeqE => t; rewrite indicE. Qed. HB.instance Definition _ (D : set aT) (mD : measurable D) : @FImFun aT rT (mindic mD) := FImFun.on (mindic mD). Lemma indic_mfun_subproof (D : set aT) (mD : measurable D) : @IsMeasurableFun d aT rT (mindic mD). Proof. split=> mA /= B mB; rewrite preimage_indic. case: ifPn => B1; case: ifPn => B0 //. - by rewrite setIT. - exact: measurableI. - by apply: measurableI => //; apply: measurableC. - by rewrite setI0. Qed. HB.instance Definition _ D mD := @indic_mfun_subproof D mD. Definition indic_mfun (D : set aT) (mD : measurable D) := [the {mfun aT >-> rT} of mindic mD]. HB.instance Definition _ k f := MeasurableFun.copy (k \o* f) (f * cst_mfun k). Definition scale_mfun k f := [the {mfun aT >-> rT} of k \o* f]. Lemma max_mfun_subproof f g : @IsMeasurableFun d aT rT (f \max g). Proof. by split; apply: measurable_fun_max. Qed. HB.instance Definition _ f g := max_mfun_subproof f g. Definition max_mfun f g := [the {mfun aT >-> _} of f \max g]. End ring. Arguments indic_mfun {d aT rT} _. Section sfun_pred. Context {d} {aT : measurableType d} {rT : realType}. Definition sfun : {pred _ -> _} := [predI @mfun _ aT rT & fimfun]. Definition sfun_key : pred_key sfun. Proof. exact. Qed. Canonical sfun_keyed := KeyedPred sfun_key. Lemma sub_sfun_mfun : {subset sfun <= mfun}. Proof. by move=> x /andP[]. Qed. Lemma sub_sfun_fimfun : {subset sfun <= fimfun}. Proof. by move=> x /andP[]. Qed. End sfun_pred. Section sfun. Context {d} {aT : measurableType d} {rT : realType}. Notation T := {sfun aT >-> rT}. Notation sfun := (@sfun _ aT rT). Section Sub. Context (f : aT -> rT) (fP : f \in sfun). Definition sfun_Sub1_subproof := @IsMeasurableFun.Build d aT rT f (set_mem (sub_sfun_mfun fP)). #[local] HB.instance Definition _ := sfun_Sub1_subproof. Definition sfun_Sub2_subproof := @FiniteImage.Build aT rT f (set_mem (sub_sfun_fimfun fP)). #[local] HB.instance Definition _ := sfun_Sub2_subproof. Definition sfun_Sub := [sfun of f]. End Sub. Lemma sfun_rect (K : T -> Type) : (forall f (Pf : f \in sfun), K (sfun_Sub Pf)) -> forall u : T, K u. Proof. move=> Ksub [f [[Pf1] [Pf2]]]; have Pf : f \in sfun by apply/andP; rewrite ?inE. have -> : Pf1 = (set_mem (sub_sfun_mfun Pf)) by []. have -> : Pf2 = (set_mem (sub_sfun_fimfun Pf)) by []. exact: Ksub. Qed. Lemma sfun_valP f (Pf : f \in sfun) : sfun_Sub Pf = f :> (_ -> _). Proof. by []. Qed. Canonical sfun_subType := SubType T _ _ sfun_rect sfun_valP. Lemma sfuneqP (f g : {sfun aT >-> rT}) : f = g <-> f =1 g. Proof. by split=> [->//|fg]; apply/val_inj/funext. Qed. Definition sfuneqMixin := [eqMixin of {sfun aT >-> rT} by <:]. Canonical sfuneqType := EqType {sfun aT >-> rT} sfuneqMixin. Definition sfunchoiceMixin := [choiceMixin of {sfun aT >-> rT} by <:]. Canonical sfunchoiceType := ChoiceType {sfun aT >-> rT} sfunchoiceMixin. (* TODO: BUG: HB *) (* HB.instance Definition _ (x : rT) := @cst_mfun_subproof aT rT x. *) Definition cst_sfun x := [the {sfun aT >-> rT} of cst x]. Lemma cst_sfunE x : @cst_sfun x =1 cst x. Proof. by []. Qed. End sfun. (* a better way to refactor function stuffs *) Lemma fctD (T : pointedType) (K : ringType) (f g : T -> K) : f + g = f \+ g. Proof. by []. Qed. Lemma fctN (T : pointedType) (K : ringType) (f : T -> K) : - f = \- f. Proof. by []. Qed. Lemma fctM (T : pointedType) (K : ringType) (f g : T -> K) : f * g = f \* g. Proof. by []. Qed. Lemma fctZ (T : pointedType) (K : ringType) (L : lmodType K) k (f : T -> L) : k *: f = k \*: f. Proof. by []. Qed. Arguments cst _ _ _ _ /. Definition fctWE := (fctD, fctN, fctM, fctZ). Section ring. Context (d : measure_display) (aT : measurableType d) (rT : realType). Lemma sfun_subring_closed : subring_closed (@sfun d aT rT). Proof. by split=> [|f g|f g]; rewrite ?inE/= ?rpred1//; move=> /andP[/= mf ff] /andP[/= mg fg]; rewrite !(rpredB, rpredM). Qed. Canonical sfun_add := AddrPred sfun_subring_closed. Canonical sfun_zmod := ZmodPred sfun_subring_closed. Canonical sfun_mul := MulrPred sfun_subring_closed. Canonical sfun_subring := SubringPred sfun_subring_closed. Definition sfun_zmodMixin := [zmodMixin of {sfun aT >-> rT} by <:]. Canonical sfun_zmodType := ZmodType {sfun aT >-> rT} sfun_zmodMixin. Definition sfun_ringMixin := [ringMixin of {sfun aT >-> rT} by <:]. Canonical sfun_ringType := RingType {sfun aT >-> rT} sfun_ringMixin. Definition sfun_comRingMixin := [comRingMixin of {sfun aT >-> rT} by <:]. Canonical sfun_comRingType := ComRingType {sfun aT >-> rT} sfun_comRingMixin. Implicit Types (f g : {sfun aT >-> rT}). Lemma sfun0 : (0 : {sfun aT >-> rT}) =1 cst 0. Proof. by []. Qed. Lemma sfun1 : (1 : {sfun aT >-> rT}) =1 cst 1. Proof. by []. Qed. Lemma sfunN f : - f =1 \- f. Proof. by []. Qed. Lemma sfunD f g : f + g =1 f \+ g. Proof. by []. Qed. Lemma sfunB f g : f - g =1 f \- g. Proof. by []. Qed. Lemma sfunM f g : f * g =1 f \* g. Proof. by []. Qed. Lemma sfun_sum I r (P : {pred I}) (f : I -> {sfun aT >-> rT}) (x : aT) : (\sum_(i <- r | P i) f i) x = \sum_(i <- r | P i) f i x. Proof. by elim/big_rec2: _ => //= i y ? Pi <-. Qed. Lemma sfun_prod I r (P : {pred I}) (f : I -> {sfun aT >-> rT}) (x : aT) : (\sum_(i <- r | P i) f i) x = \sum_(i <- r | P i) f i x. Proof. by elim/big_rec2: _ => //= i y ? Pi <-. Qed. Lemma sfunX f n : f ^+ n =1 (fun x => f x ^+ n). Proof. by move=> x; elim: n => [|n IHn]//; rewrite !exprS sfunM/= IHn. Qed. HB.instance Definition _ f g := MeasurableFun.copy (f \+ g) (f + g). HB.instance Definition _ f g := MeasurableFun.copy (\- f) (- f). HB.instance Definition _ f g := MeasurableFun.copy (f \- g) (f - g). HB.instance Definition _ f g := MeasurableFun.copy (f \* g) (f * g). Definition indic_sfun (D : set aT) (mD : measurable D) := [the {sfun aT >-> rT} of mindic rT mD]. HB.instance Definition _ k f := MeasurableFun.copy (k \o* f) (f * cst_sfun k). Definition scale_sfun k f := [the {sfun aT >-> rT} of k \o* f]. HB.instance Definition _ f g := max_mfun_subproof f g. Definition max_sfun f g := [the {sfun aT >-> _} of f \max g]. End ring. Arguments indic_sfun {d aT rT} _. Lemma fset_set_comp (T1 : Type) (T2 T3 : choiceType) (D : set T1) (f : {fimfun T1 >-> T2}) (g : T2 -> T3) : fset_set [set (g \o f) x | x in D] = [fset g x | x in fset_set [set f x | x in D]]%fset. Proof. by rewrite -(image_comp f g) fset_set_image. Qed. Lemma preimage_nnfun0 T (R : realDomainType) (f : {nnfun T >-> R}) t : t < 0 -> f @^-1` [set t] = set0. Proof. move=> t0. by apply/preimage10 => -[x _]; apply: contraPnot t0 => <-; rewrite le_gtF. Qed. Lemma preimage_cstM T (R : realFieldType) (x y : R) (f : T -> R) : x != 0 -> (cst x \* f) @^-1` [set y] = f @^-1` [set y / x]. Proof. move=> x0; apply/seteqP; rewrite /preimage; split => [z/= <-|z/= ->]. by rewrite mulrAC divrr ?mul1r// unitfE. by rewrite mulrCA divrr ?mulr1// unitfE. Qed. Lemma preimage_add T (R : numDomainType) (f g : T -> R) z : (f \+ g) @^-1` [set z] = \bigcup_(a in f @` setT) ((f @^-1` [set a]) `&` (g @^-1` [set z - a])). Proof. apply/seteqP; split=> [x /= fgz|x [_ /= [y _ <-]] []]. have : z - f x \in g @` setT. by rewrite inE /=; exists x=> //; rewrite -fgz addrC addKr. rewrite inE /= => -[x' _ gzf]; exists (z - g x')%R => /=. by exists x => //; rewrite gzf opprB addrC subrK. rewrite /preimage /=; split; first by rewrite gzf opprB addrC subrK. by rewrite gzf opprB addrC subrK -fgz addrC addKr. rewrite /preimage /= => [fxfy gzf]. by rewrite gzf -fxfy addrC subrK. Qed. Section nnsfun_functions. Variables (d : measure_display) (T : measurableType d) (R : realType). Lemma cst_nnfun_subproof (x : {nonneg R}) : @IsNonNegFun T R (cst x%:num). Proof. by split=> /=. Qed. HB.instance Definition _ x := @cst_nnfun_subproof x. Definition cst_nnsfun (r : {nonneg R}) := [the {nnsfun T >-> R} of cst r%:num]. Definition nnsfun0 : {nnsfun T >-> R} := cst_nnsfun 0%R%:nng. Lemma indic_nnfun_subproof (D : set T) : @IsNonNegFun T R (\1_D). Proof. by split=> //=; rewrite /indic. Qed. HB.instance Definition _ D := @indic_nnfun_subproof D. HB.instance Definition _ D (mD : measurable D) : @NonNegFun T R (mindic R mD) := NonNegFun.on (mindic R mD). End nnsfun_functions. Arguments nnsfun0 {d T R}. Section nnfun_bin. Variables (T : Type) (R : numDomainType) (f g : {nnfun T >-> R}). Lemma add_nnfun_subproof : @IsNonNegFun T R (f \+ g). Proof. by split => x; rewrite addr_ge0//; apply/fun_ge0. Qed. HB.instance Definition _ := add_nnfun_subproof. Lemma mul_nnfun_subproof : @IsNonNegFun T R (f \* g). Proof. by split => x; rewrite mulr_ge0//; apply/fun_ge0. Qed. HB.instance Definition _ := mul_nnfun_subproof. Lemma max_nnfun_subproof : @IsNonNegFun T R (f \max g). Proof. by split => x /=; rewrite /maxr; case: ifPn => _; apply: fun_ge0. Qed. HB.instance Definition _ := max_nnfun_subproof. End nnfun_bin. Section nnsfun_bin. Variables (d : measure_display) (T : measurableType d). Variables (R : realType) (f g : {nnsfun T >-> R}). HB.instance Definition _ := MeasurableFun.on (f \+ g). Definition add_nnsfun := [the {nnsfun T >-> R} of f \+ g]. HB.instance Definition _ := MeasurableFun.on (f \* g). Definition mul_nnsfun := [the {nnsfun T >-> R} of f \* g]. HB.instance Definition _ := MeasurableFun.on (f \max g). Definition max_nnsfun := [the {nnsfun T >-> R} of f \max g]. Definition indic_nnsfun A (mA : measurable A) := [the {nnsfun T >-> R} of mindic R mA]. End nnsfun_bin. Arguments add_nnsfun {d T R} _ _. Arguments mul_nnsfun {d T R} _ _. Arguments max_nnsfun {d T R} _ _. Section nnsfun_iter. Variables (d : measure_display) (T : measurableType d) (R : realType) (D : set T). Variable f : {nnsfun T >-> R}^nat. Definition sum_nnsfun n := \big[add_nnsfun/nnsfun0]_(i < n) f i. Lemma sum_nnsfunE n t : sum_nnsfun n t = \sum_(i < n) (f i t). Proof. by rewrite /sum_nnsfun; elim/big_ind2 : _ => [|x g y h <- <-|]. Qed. Definition bigmax_nnsfun n := \big[max_nnsfun/nnsfun0]_(i < n) f i. Lemma bigmax_nnsfunE n t : bigmax_nnsfun n t = \big[maxr/0]_(i < n) (f i t). Proof. by rewrite /bigmax_nnsfun; elim/big_ind2 : _ => [|x g y h <- <-|]. Qed. End nnsfun_iter. Section nnsfun_cover. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d). Variables (R : realType) (f : {nnsfun T >-> R}). Lemma nnsfun_cover : \big[setU/set0]_(i \in range f) (f @^-1` [set i]) = setT. Proof. by rewrite fsbig_setU//= -subTset => x _; exists (f x). Qed. Lemma nnsfun_coverT : \big[setU/set0]_(i \in [set: R]) (f @^-1` [set i]) = setT. Proof. by rewrite -(fsbig_widen (range f)) ?nnsfun_cover//= => x [_ /= /preimage10->]. Qed. End nnsfun_cover. #[global] Hint Extern 0 (measurable (_ @^-1` [set _])) => solve [apply: measurable_sfunP] : core. Lemma measurable_sfun_inP {d} {aT : measurableType d} {rT : realType} (f : {mfun aT >-> rT}) D (y : rT) : measurable D -> measurable (D `&` f @^-1` [set y]). Proof. by move=> Dm; apply: measurableI. Qed. #[global] Hint Extern 0 (measurable (_ `&` _ @^-1` [set _])) => solve [apply: measurable_sfun_inP; assumption] : core. #[global] Hint Extern 0 (finite_set _) => solve [apply: fimfunP] : core. Section measure_fsbig. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d). Variables (R : realType) (m : {measure set T -> \bar R}). Lemma measure_fsbig (I : choiceType) (A : set I) (F : I -> set T) : finite_set A -> (forall i, A i -> measurable (F i)) -> trivIset A F -> m (\big[setU/set0]_(i \in A) F i) = \sum_(i \in A) m (F i). Proof. move=> Afin Fm Ft. by rewrite fsbig_finite// -measure_fin_bigcup// bigcup_fset_set. Qed. Lemma additive_nnsfunr (g f : {nnsfun T >-> R}) x : \sum_(i \in range g) m (f @^-1` [set x] `&` (g @^-1` [set i])) = m (f @^-1` [set x] `&` \big[setU/set0]_(i \in range g) (g @^-1` [set i])). Proof. rewrite -?measure_fsbig//. - by rewrite !fsbig_finite//= big_distrr//. - by move=> i Ai; apply: measurableI => //. - exact/trivIset_setI/trivIset_preimage1. Qed. Lemma additive_nnsfunl (g f : {nnsfun T >-> R}) x : \sum_(i \in range g) m (g @^-1` [set i] `&` (f @^-1` [set x])) = m (\big[setU/set0]_(i \in range g) (g @^-1` [set i]) `&` f @^-1` [set x]). Proof. by under eq_fsbigr do rewrite setIC; rewrite setIC additive_nnsfunr. Qed. End measure_fsbig. Section mulem_ge0. Local Open Scope ereal_scope. Let mulef_ge0 (R : realDomainType) x (f : R -> \bar R) : (forall x, 0 <= f x) -> ((x < 0)%R -> f x = 0) -> 0 <= x%:E * f x. Proof. move=> A0 xA /=; have [x0|x0] := ltP x 0%R; first by rewrite (xA x0) mule0. by rewrite mule_ge0. Qed. Lemma muleindic_ge0 d (T : measurableType d) (R : realDomainType) (f : {nnfun T >-> R}) r z : 0 <= r%:E * (\1_(f @^-1` [set r]) z)%:E. Proof. apply: (@mulef_ge0 _ _ (fun r => (\1_(f @^-1` [set r]) z)%:E)). by move=> x; rewrite lee_fin /indic. by move=> r0; rewrite preimage_nnfun0// indic0. Qed. Lemma mulem_ge0 d (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}) x (A : R -> set T) : ((x < 0)%R -> A x = set0) -> 0 <= x%:E * mu (A x). Proof. by move=> xA; rewrite (@mulef_ge0 _ _ (mu \o _))//= => /xA ->; rewrite measure0. Qed. Arguments mulem_ge0 {d T R mu x} A. Lemma nnfun_mulem_ge0 d (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R})(f : {nnsfun T >-> R}) x : 0 <= x%:E * mu (f @^-1` [set x]). Proof. by apply: (mulem_ge0 (fun x => f @^-1` [set x])); exact: preimage_nnfun0. Qed. End mulem_ge0. (**********************************) (* Definition of Simple Integrals *) (**********************************) Section simple_fun_raw_integral. Local Open Scope ereal_scope. Variables (T : Type) (R : numDomainType) (mu : set T -> \bar R) (f : T -> R). Definition sintegral := \sum_(x \in [set: R]) x%:E * mu (f @^-1` [set x]). Lemma sintegralET : sintegral = \sum_(x \in [set: R]) x%:E * mu (f @^-1` [set x]). Proof. by []. Qed. End simple_fun_raw_integral. #[global] Hint Extern 0 (is_true (0 <= (_ : {measure set _ -> \bar _}) _)%E) => solve [apply: measure_ge0] : core. Section sintegral_lemmas. Variables (d : measure_display) (T : measurableType d). Variables (R : realType) (mu : {measure set T -> \bar R}). Local Open Scope ereal_scope. Lemma sintegralE f : sintegral mu f = \sum_(x \in range f) x%:E * mu (f @^-1` [set x]). Proof. rewrite (fsbig_widen (range f) setT)//= => x [_ Nfx] /=. by rewrite preimage10// measure0 mule0. Qed. Lemma sintegral0 : sintegral mu (cst 0%R) = 0. Proof. rewrite sintegralE fsbig1// => r _; rewrite preimage_cst. by case: ifPn => [/[!inE] <-|]; rewrite ?mul0e// measure0 mule0. Qed. Lemma sintegral_ge0 (f : {nnsfun T >-> R}) : 0 <= sintegral mu f. Proof. by rewrite sintegralE fsume_ge0// => r _; exact: nnfun_mulem_ge0. Qed. Lemma sintegral_indic (A : set T) : sintegral mu \1_A = mu A. Proof. rewrite sintegralE (fsbig_widen _ [set 0%R; 1%R]) => //; last 2 first. - exact: image_indic_sub. - by move=> t [[] -> /= /preimage10->]; rewrite measure0 mule0. have N01 : (0 <> 1:> R)%R by move=> /esym/eqP; rewrite oner_eq0. rewrite fsbigU//=; last by move=> t [->]//. rewrite !fsbig_set1 mul0e add0e mul1e. by rewrite preimage_indic ifT ?inE// ifN ?notin_set. Qed. (* NB: not used *) Lemma sintegralEnnsfun (f : {nnsfun T >-> R}) : sintegral mu f = (\sum_(x \in [set r | r > 0]%R) (x%:E * mu (f @^-1` [set x])))%E. Proof. rewrite (fsbig_widen _ setT) ?sintegralET//. move=> x [_ /=]; case: ltgtP => //= [xlt0 _|<-]; last by rewrite mul0e. rewrite preimage10 ?measure0 ?mule0//= => -[t _]. by apply/eqP; apply: contra_ltN xlt0 => /eqP<-. Qed. End sintegral_lemmas. Lemma eq_sintegral d (T : measurableType d) (R : numDomainType) (mu : set T -> \bar R) g f : f =1 g -> sintegral mu f = sintegral mu g. Proof. by move=> /funext->. Qed. Arguments eq_sintegral {d T R mu} g. Section sintegralrM. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d). Variables (R : realType) (m : {measure set T -> \bar R}). Variables (r : R) (f : {nnsfun T >-> R}). Lemma sintegralrM : sintegral m (cst r \* f)%R = r%:E * sintegral m f. Proof. have [->|r0] := eqVneq r 0%R. by rewrite mul0e (eq_sintegral (cst 0%R)) ?sintegral0// => x/=; rewrite mul0r. rewrite !sintegralET. transitivity (\sum_(x \in [set: R]) x%:E * m (f @^-1` [set x / r])). by apply: eq_fsbigr => x; rewrite preimage_cstM. transitivity (\sum_(x \in [set: R]) r%:E * (x%:E * m (f @^-1` [set x]))). rewrite (reindex_fsbigT (fun x => r * x)%R)//; last first. by exists ( *%R r ^-1)%R; [exact: mulKf|exact: mulVKf]. by apply: eq_fsbigr => x; rewrite mulrAC divrr ?unitfE// mul1r muleA EFinM. by rewrite ge0_mule_fsumr// => x; exact: nnfun_mulem_ge0. Qed. End sintegralrM. Section sintegralD. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). Variables (m : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D) (f g : {nnsfun T >-> R}). Lemma sintegralD : sintegral m (f \+ g)%R = sintegral m f + sintegral m g. Proof. rewrite !sintegralE; set F := f @` _; set G := g @` _; set FG := _ @` _. pose pf x := f @^-1` [set x]; pose pg y := g @^-1` [set y]. transitivity (\sum_(z \in FG) z%:E * \sum_(a \in F) m (pf a `&` pg (z - a)%R)). apply: eq_fsbigr => z _; rewrite preimage_add -fsbig_setU// measure_fsbig//. by move=> x Fx; apply: measurableI. exact/trivIset_setIr/trivIset_preimage1. under eq_fsbigr do rewrite ge0_mule_fsumr//; rewrite exchange_fsum//. transitivity (\sum_(x \in F) \sum_(y \in G) (x + y)%:E * m (pf x `&` pg y)). apply: eq_fsbigr => x _; rewrite /pf /pg (fsbig_widen G setT)//=; last first. by move=> y [_ /= /preimage10->]; rewrite setI0 measure0 mule0. rewrite (fsbig_widen FG setT)//=; last first. move=> z [_ /= FGz]; rewrite [X in m X](_ : _ = set0) ?measure0 ?mule0//. rewrite -subset0 => //= {x}i /= [<-] /(canLR (@addrNK _ _)). by apply: contra_not FGz => <-; exists i; rewrite //= addrC. rewrite (reindex_fsbigT (+%R x))//=. by apply: eq_fsbigr => y; rewrite addrC addrK. transitivity (\sum_(x \in F) \sum_(y \in G) x%:E * m (pf x `&` pg y) + \sum_(x \in F) \sum_(y \in G) y%:E * m (pf x `&` pg y)). do 2![rewrite -fsbig_split//; apply: eq_fsbigr => _ /set_mem [? _ <-]]. by rewrite EFinD ge0_muleDl// ?lee_fin. congr (_ + _)%E; last rewrite exchange_fsum//; apply: eq_fsbigr => x _. by rewrite -ge0_mule_fsumr// additive_nnsfunr nnsfun_cover setIT. by rewrite -ge0_mule_fsumr// additive_nnsfunl nnsfun_cover setTI. Qed. End sintegralD. Section le_sintegral. Variables (d : measure_display) (T : measurableType d). Variables (R : realType) (m : {measure set T -> \bar R}). Variables f g : {nnsfun T >-> R}. Hypothesis fg : forall x, f x <= g x. Let fgnn : @IsNonNegFun T R (g \- f). Proof. by split=> x; rewrite subr_ge0 fg. Qed. #[local] HB.instance Definition _ := fgnn. Lemma le_sintegral : (sintegral m f <= sintegral m g)%E. Proof. have gfgf : g =1 f \+ (g \- f) by move=> x /=; rewrite addrC subrK. by rewrite (eq_sintegral _ _ gfgf) sintegralD// lee_addl // sintegral_ge0. Qed. End le_sintegral. Lemma is_cvg_sintegral d (T : measurableType d) (R : realType) (m : {measure set T -> \bar R}) (f : {nnsfun T >-> R}^nat) : (forall x, nondecreasing_seq (f ^~ x)) -> cvg (sintegral m \o f). Proof. move=> nd_f; apply/cvg_ex; eexists; apply/ereal_nondecreasing_cvg => a b ab. by apply: le_sintegral => // => x; exact/nd_f. Qed. Definition proj_nnsfun d (T : measurableType d) (R : realType) (f : {nnsfun T >-> R}) (A : set T) (mA : measurable A) := mul_nnsfun f (indic_nnsfun R mA). Definition mrestrict d (T : measurableType d) (R : realType) (f : {nnsfun T >-> R}) A (mA : measurable A) : f \_ A = proj_nnsfun f mA. Proof. apply/funext => x /=; rewrite /patch mindicE. by case: ifP; rewrite (mulr0, mulr1). Qed. Definition scale_nnsfun d (T : measurableType d) (R : realType) (f : {nnsfun T >-> R}) (k : R) (k0 : 0 <= k) := mul_nnsfun (cst_nnsfun T (NngNum k0)) f. Section sintegral_nondecreasing_limit_lemma. Variables (d : measure_display) (T : measurableType d) (R : realType). Variables (mu : {measure set T -> \bar R}). Variables (g : {nnsfun T >-> R}^nat) (f : {nnsfun T >-> R}). Hypothesis nd_g : forall x, nondecreasing_seq (g^~ x). Hypothesis gf : forall x, cvg (g^~ x) -> f x <= lim (g^~ x). Let fleg c : (set T)^nat := fun n => [set x | c * f x <= g n x]. Let nd_fleg c : {homo fleg c : n m / (n <= m)%N >-> (n <= m)%O}. Proof. move=> n m nm; rewrite /fleg; apply/subsetPset => x /= cfg. by move: cfg => /le_trans; apply; exact: nd_g. Qed. Let mfleg c n : measurable (fleg c n). Proof. rewrite /fleg [X in _ X](_ : _ = \big[setU/set0]_(y <- fset_set (range f)) \big[setU/set0]_(x <- fset_set (range (g n)) | c * y <= x) (f @^-1` [set y] `&` (g n @^-1` [set x]))). apply: bigsetU_measurable => r _; apply: bigsetU_measurable => r' crr'. by apply: measurableI; apply/measurable_sfunP. rewrite predeqE => t; split => [/= cfgn|]. - rewrite -bigcup_set; exists (f t); first by rewrite /= in_fset_set//= mem_set. rewrite -bigcup_set_cond; exists (g n t) => //=. by rewrite in_fset_set// mem_set. - rewrite -bigcup_fset_set// => -[r [x _ fxr]]. rewrite -bigcup_fset_set_cond// => -[r' [[x' _ gnx'r'] crr']]. by rewrite /preimage/= => -[-> ->]. Qed. Let g1 c n : {nnsfun T >-> R} := proj_nnsfun f (mfleg c n). Let le_ffleg c : {homo (fun p x => g1 c p x): m n / (m <= n)%N >-> (m <= n)%O}. Proof. move=> m n mn; apply/asboolP => t; rewrite /g1/= ler_pmul// 2!mindicE/= ler_nat. have [|//] := boolP (t \in fleg c m); rewrite inE => cnt. by have := nd_fleg c mn => /subsetPset/(_ _ cnt) cmt; rewrite mem_set. Qed. Let bigcup_fleg c : c < 1 -> \bigcup_n fleg c n = setT. Proof. move=> c1; rewrite predeqE => x; split=> // _. have := @fun_ge0 _ _ f x; rewrite le_eqVlt => /predU1P[|] gx0. by exists O => //; rewrite /fleg /=; rewrite -gx0 mulr0 fun_ge0. have [cf|df] := pselect (cvg (g^~ x)). have cfg : lim (g^~ x) > c * f x. by rewrite (lt_le_trans _ (gf cf)) // gtr_pmull. suff [n cfgn] : exists n, g n x >= c * f x by exists n. move/(@lt_lim _ _ _ (nd_g x) cf) : cfg => [n _ nf]. by exists n; apply: nf => /=. have /cvgPpinfty/(_ (c * f x))[n _ ncfgn]:= nondecreasing_dvg_lt (nd_g x) df. by exists n => //; rewrite /fleg /=; apply: ncfgn => /=. Qed. Local Open Scope ereal_scope. Lemma nd_sintegral_lim_lemma : sintegral mu f <= lim (sintegral mu \o g). Proof. suff ? : forall c, (0 < c < 1)%R -> c%:E * sintegral mu f <= lim (sintegral mu \o g). by apply/lee_mul01Pr => //; exact: sintegral_ge0. move=> c /andP[c0 c1]. have cg1g n : c%:E * sintegral mu (g1 c n) <= sintegral mu (g n). rewrite -sintegralrM (_ : (_ \* _)%R = scale_nnsfun (g1 c n) (ltW c0)) //. apply: le_sintegral => // t. suff : forall m x, (c * g1 c m x <= g m x)%R by move=> /(_ n t). move=> m x; rewrite /g1 /proj_nnsfun/= mindicE. by have [|] := boolP (_ \in _); [rewrite inE mulr1|rewrite 2!mulr0 fun_ge0]. suff {cg1g}<- : lim (fun n => sintegral mu (g1 c n)) = sintegral mu f. have is_cvg_g1 : cvg (fun n => sintegral mu (g1 c n)). by apply: is_cvg_sintegral => //= x m n /(le_ffleg c)/lefP/(_ x). rewrite -ereal_limrM // lee_lim//; first exact: ereal_is_cvgrM. - by apply: is_cvg_sintegral => // m n mn; apply/lefP => t; apply: nd_g. - by apply: nearW; exact: cg1g. suff : (fun n => sintegral mu (g1 c n)) --> sintegral mu f by apply/cvg_lim. rewrite [X in X --> _](_ : _ = fun n => \sum_(x <- fset_set (range f)) x%:E * mu (f @^-1` [set x] `&` fleg c n)); last first. rewrite funeqE => n; rewrite sintegralE. transitivity (\sum_(x \in range f) x%:E * mu (g1 c n @^-1` [set x])). apply: eq_fbigl => r. do 2 (rewrite in_finite_support; last exact/finite_setIl). apply/idP/idP. rewrite in_setI => /andP[]; rewrite inE/= => -[x _]; rewrite mindicE. have [_|xcn] := boolP (_ \in _). by rewrite mulr1 => <-; rewrite !inE/= => ?; split => //; exists x. by rewrite mulr0 => /esym ->; rewrite !inE/= mul0e. rewrite in_setI => /andP[]; rewrite inE => -[x _ <-]. rewrite !inE/= => h; split=> //; move: h; rewrite mindicE => /eqP. rewrite mule_eq0 negb_or => /andP[_]; set S := (X in mu X) => mS0. suff : S !=set0 by move=> [y yx]; exists y. by apply/set0P; apply: contra mS0 => /eqP ->; rewrite measure0. rewrite fsbig_finite//=; apply: eq_fbigr => r. rewrite in_fset_set// inE => -[t _ ftr _]. have [->|r0] := eqVneq r 0%R; first by rewrite 2!mul0e. congr (_ * mu _); apply/seteqP; split => x. rewrite /preimage/= mindicE. have [|_] := boolP (_ \in _); first by rewrite mulr1 inE. by rewrite mulr0 => /esym/eqP; rewrite (negbTE r0). by rewrite /preimage/= => -[fxr cnx]; rewrite mindicE mem_set// mulr1. rewrite sintegralE fsbig_finite//=; apply: ereal_lim_sum => [r n _|r _]. apply: (@mulem_ge0 _ _ _ _ _ (fun x => f @^-1` [set x] `&` fleg c n)) => r0. by rewrite preimage_nnfun0// set0I. apply: ereal_cvgrM => //; rewrite [X in _ --> X](_ : _ = mu (\bigcup_n (f @^-1` [set r] `&` fleg c n))); last first. by rewrite -setI_bigcupr bigcup_fleg// setIT. have ? k i : measurable (f @^-1` [set k] `&` fleg c i) by exact: measurableI. apply: cvg_mu_inc => //; first exact: bigcupT_measurable. move=> n m nm; apply/subsetPset; apply: setIS. by move/(nd_fleg c) : nm => /subsetPset. Unshelve. all: by end_near. Qed. End sintegral_nondecreasing_limit_lemma. Section sintegral_nondecreasing_limit. Variables (d : measure_display) (T : measurableType d) (R : realType). Variables (mu : {measure set T -> \bar R}). Variables (g : {nnsfun T >-> R}^nat) (f : {nnsfun T >-> R}). Hypothesis nd_g : forall x, nondecreasing_seq (g^~ x). Hypothesis gf : forall x, g ^~ x --> f x. Let limg x : lim (g^~x) = f x. Proof. by apply/cvg_lim; [exact: Rhausdorff| exact: gf]. Qed. Lemma nd_sintegral_lim : sintegral mu f = lim (sintegral mu \o g). Proof. apply/eqP; rewrite eq_le; apply/andP; split. by apply: nd_sintegral_lim_lemma => // x; rewrite -limg. have : nondecreasing_seq (sintegral mu \o g). by move=> m n mn; apply: le_sintegral => // x; exact/nd_g. move=> /ereal_nondecreasing_cvg/cvg_lim -> //. apply: ub_ereal_sup => _ [n _ <-] /=; apply: le_sintegral => // x. rewrite -limg // (nondecreasing_cvg_le (nd_g x)) //. by apply/cvg_ex; exists (f x); exact: gf. Qed. End sintegral_nondecreasing_limit. Section integral. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). Implicit Types (f g : T -> \bar R) (D : set T). Let nnintegral mu f := ereal_sup [set sintegral mu h | h in [set h : {nnsfun T >-> R} | forall x, (h x)%:E <= f x]]. Definition integral mu D f (g := f \_ D) := nnintegral mu (g ^\+) - nnintegral mu (g ^\-). Variable (mu : {measure set T -> \bar R}). Let nnintegral_ge0 f : (forall x, 0 <= f x) -> 0 <= nnintegral mu f. Proof. by move=> f0; apply: ereal_sup_ub; exists nnsfun0; last by rewrite sintegral0. Qed. Let eq_nnintegral g f : f =1 g -> nnintegral mu f = nnintegral mu g. Proof. by move=> /funext->. Qed. Let nnintegral0 : nnintegral mu (cst 0) = 0. Proof. rewrite /nnintegral /=; apply/eqP; rewrite eq_le; apply/andP; split; last first. apply/ereal_sup_ub; exists nnsfun0; last by rewrite sintegral0. by []. apply/ub_ereal_sup => /= x [f /= f0 <-]; have {}f0 : forall x, f x = 0%R. by move=> y; apply/eqP; rewrite eq_le -2!lee_fin f0 //= lee_fin//. by rewrite (eq_sintegral (@nnsfun0 _ T R)) ?sintegral0. Qed. Let nnintegral_nnsfun (h : {nnsfun T >-> R}) : nnintegral mu (EFin \o h) = sintegral mu h. Proof. apply/eqP; rewrite eq_le; apply/andP; split. by apply/ub_ereal_sup => /= _ -[g /= gh <-]; rewrite le_sintegral. by apply: ereal_sup_ub => /=; exists h. Qed. Local Notation "\int_ ( x 'in' D ) F" := (integral mu D (fun x => F)) (at level 36, F at level 36, x, D at level 50, format "'[' \int_ ( x 'in' D ) '/ ' F ']'"). Lemma eq_integral D g f : {in D, f =1 g} -> \int_(x in D) f x = \int_(x in D) g x. Proof. by rewrite /integral => /eq_restrictP->. Qed. Lemma ge0_integralE D f : (forall x, D x -> 0 <= f x) -> \int_(x in D) f x = nnintegral mu (f \_ D). Proof. move=> f0; rewrite /integral funeneg_restrict funepos_restrict. have /eq_restrictP-> := ge0_funeposE f0. have /eq_restrictP-> := ge0_funenegE f0. by rewrite erestrict0 nnintegral0 sube0. Qed. Lemma ge0_integralTE f : (forall x, 0 <= f x) -> \int_(x in setT) f x = nnintegral mu f. Proof. by move=> f0; rewrite ge0_integralE// patch_setT. Qed. Lemma integralE D f : \int_(x in D) f x = \int_(x in D) (f ^\+ x) - \int_(x in D) f ^\- x. Proof. by rewrite [in LHS]/integral funepos_restrict funeneg_restrict -!ge0_integralE. Qed. Lemma integral0 D : \int_(x in D) (cst 0 x) = 0. Proof. by rewrite ge0_integralE// erestrict0 nnintegral0. Qed. Lemma integral_ge0 D f : (forall x, D x -> 0 <= f x) -> 0 <= \int_(x in D) f x. Proof. move=> f0; rewrite ge0_integralE// nnintegral_ge0// => x. by rewrite /patch; case: ifP; rewrite // inE => /f0->. Qed. Lemma integral_nnsfun D (mD : measurable D) (h : {nnsfun T >-> R}) : \int_(x in D) (h x)%:E = sintegral mu (h \_ D). Proof. rewrite mrestrict -nnintegral_nnsfun// -mrestrict ge0_integralE ?comp_patch//. by move=> x Dx /=; rewrite lee_fin; exact: fun_ge0. Qed. End integral. Notation "\int [ mu ]_ ( x 'in' D ) f" := (integral mu D (fun x => f)) : ereal_scope. Notation "\int [ mu ]_ x f" := ((integral mu setT (fun x => f)))%E : ereal_scope. Arguments eq_integral {d T R mu D} g. Section eq_measure_integral. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (D : set T). Implicit Types m : {measure set T -> \bar R}. Let eq_measure_integral0 m2 m1 (f : T -> \bar R) : (forall A, measurable A -> A `<=` D -> m1 A = m2 A) -> [set sintegral m1 h | h in [set h : {nnsfun T >-> R} | (forall x, (h x)%:E <= (f \_ D) x)]] `<=` [set sintegral m2 h | h in [set h : {nnsfun T >-> R} | (forall x, (h x)%:E <= (f \_ D) x)]]. Proof. move=> m12 _ [h hfD <-] /=; exists h => //; apply: eq_fsbigr => r _. have [hrD|hrD] := pselect (h @^-1` [set r] `<=` D); first by rewrite m12. suff : r = 0%R by move=> ->; rewrite !mul0e. apply: contra_notP hrD => /eqP r0 t/= htr. have := hfD t. rewrite /patch/=; case: ifPn; first by rewrite inE. move=> tD. move: r0; rewrite -htr => ht0. by rewrite le_eqVlt eqe (negbTE ht0)/= lte_fin// ltNge// fun_ge0. Qed. Lemma eq_measure_integral m2 m1 (f : T -> \bar R) : (forall A, measurable A -> A `<=` D -> m1 A = m2 A) -> \int[m1]_(x in D) f x = \int[m2]_(x in D) f x. Proof. move=> m12; rewrite /integral funepos_restrict funeneg_restrict. by congr (ereal_sup _ - ereal_sup _)%E; rewrite eqEsubset; split; apply: eq_measure_integral0 => A /m12 // /[apply]. Qed. End eq_measure_integral. Arguments eq_measure_integral {d T R D} m2 {m1 f}. Section integral_measure_zero. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). Let sintegral_measure_zero (f : T -> R) : sintegral mzero f = 0. Proof. by rewrite sintegralE big1// => r _ /=; rewrite /mzero mule0. Qed. Lemma integral_measure_zero (D : set T) (f : T -> \bar R) : \int[mzero]_(x in D) f x = 0. Proof. have h g : (forall x, 0 <= g x) -> [set sintegral mzero h | h in [set h : {nnsfun T >-> R} | forall x, (h x)%:E <= g x]] = [set 0]. move=> g0; apply/seteqP; split => [_ [h/= Dt <-]|x -> /=]. by rewrite sintegral_measure_zero. by exists (cst_nnsfun _ (@NngNum _ 0 (lexx _))). rewrite integralE !ge0_integralE//= h ?ereal_sup1; last first. by move=> r; rewrite erestrict_ge0. by rewrite h ?ereal_sup1 ?subee// => r; rewrite erestrict_ge0. Qed. End integral_measure_zero. Section domain_change. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Lemma integral_mkcond D f : \int[mu]_(x in D) f x = \int[mu]_x (f \_ D) x. Proof. by rewrite /integral patch_setT. Qed. Lemma integralT_nnsfun (h : {nnsfun T >-> R}) : \int[mu]_x (h x)%:E = sintegral mu h. Proof. by rewrite integral_nnsfun// patch_setT. Qed. Lemma integral_mkcondr D P f : \int[mu]_(x in D `&` P) f x = \int[mu]_(x in D) (f \_ P) x. Proof. by rewrite integral_mkcond [RHS]integral_mkcond patch_setI. Qed. Lemma integral_mkcondl D P f : \int[mu]_(x in P `&` D) f x = \int[mu]_(x in D) (f \_ P) x. Proof. by rewrite setIC integral_mkcondr. Qed. End domain_change. Arguments integral_mkcond {d T R mu} D f. Section nondecreasing_integral_limit. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). Variables (mu : {measure set T -> \bar R}) (f : T -> \bar R) (g : {nnsfun T >-> R}^nat). Hypothesis f0 : forall x, 0 <= f x. Hypothesis mf : measurable_fun setT f. Hypothesis nd_g : forall x, nondecreasing_seq (g^~x). Hypothesis gf : forall x, EFin \o g^~x --> f x. Local Open Scope ereal_scope. Lemma nd_ge0_integral_lim : \int[mu]_x f x = lim (sintegral mu \o g). Proof. rewrite ge0_integralTE//. apply/eqP; rewrite eq_le; apply/andP; split; last first. apply: ereal_lim_le; first exact: is_cvg_sintegral. near=> n; apply: ereal_sup_ub; exists (g n) => //= => x. have <- : lim (EFin \o g ^~ x) = f x by apply/cvg_lim => //; exact: gf. have : (EFin \o g ^~ x) --> ereal_sup (range (EFin \o g ^~ x)). by apply: ereal_nondecreasing_cvg => p q pq /=; rewrite lee_fin; exact/nd_g. by move/cvg_lim => -> //; apply: ereal_sup_ub; exists n. have := leey (\int[mu]_x (f x)). rewrite le_eqVlt => /predU1P[|] mufoo; last first. have : \int[mu]_x (f x) \is a fin_num. by rewrite ge0_fin_numE//; exact: integral_ge0. rewrite ge0_integralTE// => /ub_ereal_sup_adherent h. apply: lee_adde => e; have {h} [/= _ [G Gf <-]] := h _ [gt0 of e%:num]. rewrite EFinN lte_subl_addr// => fGe. have : forall x, cvg (g^~ x) -> (G x <= lim (g ^~ x))%R. move=> x cg; rewrite -lee_fin -(EFin_lim cg). by have /cvg_lim gxfx := @gf x; rewrite (le_trans (Gf _))// gxfx. move=> /(nd_sintegral_lim_lemma mu nd_g)/(lee_add2r e%:num%:E). by apply: le_trans; exact: ltW. suff : lim (sintegral mu \o g) = +oo. by move=> ->; rewrite -ge0_integralTE// mufoo. apply/eq_pinftyP => r r0. have [G [Gf rG]] : exists h : {nnsfun T >-> R}, (forall x, (h x)%:E <= f x) /\ (r%:E <= sintegral mu h). have : r%:E < \int[mu]_x (f x). move: (mufoo) => /eq_pinftyP/(_ _ (addr_gt0 r0 r0)). by apply: lt_le_trans => //; rewrite lte_fin ltr_addr. rewrite ge0_integralTE// => /ereal_sup_gt[x [/= G Gf Gx rx]]. by exists G; split => //; rewrite (le_trans (ltW rx)) // Gx. have : forall x, cvg (g^~ x) -> (G x <= lim (g^~ x))%R. move=> x cg; rewrite -lee_fin -(EFin_lim cg). by have /cvg_lim gxfx := @gf x; rewrite (le_trans (Gf _)) // gxfx. by move/(nd_sintegral_lim_lemma mu nd_g) => Gg; rewrite (le_trans rG). Unshelve. all: by end_near. Qed. End nondecreasing_integral_limit. Section dyadic_interval. Variable R : realType. Definition dyadic_itv n k : interval R := `[(k%:R * 2 ^- n), (k.+1%:R * 2 ^- n)[. Local Notation I := dyadic_itv. Lemma dyadic_itv_subU n k : [set` I n k] `<=` [set` I n.+1 k.*2] `|` [set` I n.+1 k.*2.+1]. Proof. move=> r /=; rewrite in_itv /= => /andP[Ir rI]. have [rk|rk] := ltP r (k.*2.+1%:R * (2%:R ^- n.+1)); [left|right]. - rewrite in_itv /= rk andbT (le_trans _ Ir)// -muln2. rewrite natrM exprS invrM ?unitfE// ?expf_neq0// -mulrA (mulrCA 2). by rewrite divrr ?unitfE// mulr1. - rewrite in_itv /= rk /= (lt_le_trans rI)// -doubleS. rewrite -muln2 natrM exprS invrM ?unitfE// ?expf_neq0// -mulrA (mulrCA 2). by rewrite divrr ?unitfE// mulr1. Qed. Lemma bigsetU_dyadic_itv n : `[n%:R, n.+1%:R[%classic = \big[setU/set0]_(n * 2 ^ n.+1 <= k < n.+1 * 2 ^ n.+1) [set` I n.+1 k]. Proof. rewrite predeqE => r; split => [/= /[!in_itv]/= /andP[nr rn1]|]. - rewrite -bigcup_set /=; exists `|floor (r * 2 ^+ n.+1)|%N. rewrite /= mem_index_iota; apply/andP; split. rewrite -ltez_nat gez0_abs ?floor_ge0; last first. by rewrite mulr_ge0// (le_trans _ nr). apply: (@le_trans _ _ (floor (n * 2 ^ n.+1)%:R)); last first. by apply: le_floor; rewrite natrM natrX ler_pmul2r. by rewrite floor_natz intz. rewrite -ltz_nat gez0_abs; last first. by rewrite floor_ge0 mulr_ge0// (le_trans _ nr). rewrite -(@ltr_int R) (le_lt_trans (floor_le _))//. by rewrite PoszM intrM -natrX ltr_pmul2r. rewrite /= in_itv /=; apply/andP; split. rewrite ler_pdivr_mulr// (le_trans _ (floor_le _))//. by rewrite -(@gez0_abs (floor _))// floor_ge0 mulr_ge0// (le_trans _ nr). rewrite ltr_pdivl_mulr// (lt_le_trans (lt_succ_Rfloor _))// RfloorE. rewrite -[in leRHS]addn1 natrD ler_add2r// -(@gez0_abs (floor _))// floor_ge0. by rewrite mulr_ge0// (le_trans _ nr). - rewrite -bigcup_set => -[/= k] /[!mem_index_iota] /andP[nk kn]. rewrite in_itv /= => /andP[knr rkn]; rewrite in_itv /=; apply/andP; split. by rewrite (le_trans _ knr)// ler_pdivl_mulr// -natrX -natrM ler_nat. by rewrite (lt_le_trans rkn)// ler_pdivr_mulr// -natrX -natrM ler_nat. Qed. Lemma dyadic_itv_image n T (f : T -> \bar R) x : (n%:R%:E <= f x < n.+1%:R%:E)%E -> exists k, (2 ^ n.+1 * n <= k < 2 ^ n.+1 * n.+1)%N /\ f x \in EFin @` [set` I n.+1 k]. Proof. move=> fxn; have fxfin : f x \is a fin_num. by rewrite fin_numE; move: fxn; case: (f x) => // /andP[]. have : f x \in EFin @` `[n%:R, n.+1%:R[%classic. rewrite inE /=; exists (fine (f x)); last by rewrite fineK. by rewrite in_itv /= -lee_fin -lte_fin (fineK fxfin). rewrite (bigsetU_dyadic_itv n) inE /= => -[r]; rewrite -bigcup_set => -[k /=]. rewrite mem_index_iota => nk Ir rfx. by exists k; split; [rewrite !(mulnC (2 ^ n.+1)%N)|rewrite !inE /=; exists r]. Qed. End dyadic_interval. Section approximation. Variables (d : measure_display) (T : measurableType d) (R : realType). Variables (D : set T) (mD : measurable D). Variables (f : T -> \bar R) (mf : measurable_fun D f). Local Notation I := (@dyadic_itv R). Let A n k := if (k < n * 2 ^ n)%N then D `&` [set x | f x \in EFin @` [set` I n k]] else set0. Let B n := D `&` [set x | n%:R%:E <= f x]%E. Definition approx : (T -> R)^nat := fun n x => \sum_(k < n * 2 ^ n) k%:R * 2 ^- n * (x \in A n k)%:R + n%:R * (x \in B n)%:R. (* technical properties of the sets A and B *) Let mA n k : measurable (A n k). Proof. rewrite /A; case: ifPn => [kn|_]//; rewrite -preimage_comp. by apply: mf => //; apply/measurable_EFin; apply: measurable_itv. Qed. Let trivIsetA n : trivIset setT (A n). Proof. apply/trivIsetP => i j _ _. wlog : i j / (i < j)%N. move=> h; rewrite neq_lt => /orP[ij|ji]. by apply: h => //; rewrite lt_eqF. by rewrite setIC; apply: h => //; rewrite lt_eqF. move=> ij _. rewrite /A; case: ifPn => /= ni; last by rewrite set0I. case: ifPn => /= nj; last by rewrite setI0. rewrite predeqE => t; split => // -[/=] [_]. rewrite inE => -[r /=]; rewrite in_itv /= => /andP[r1 r2] rft [_]. rewrite inE => -[s /=]; rewrite in_itv /= => /andP[s1 s2]. rewrite -rft => -[sr]; rewrite {}sr {s} in s1 s2. have := le_lt_trans s1 r2. by rewrite ltr_pmul2r// ltr_nat ltnS leqNgt ij. Qed. Let f0_A0 n (i : 'I_(n * 2 ^ n)) x : f x = 0%:E -> i != O :> nat -> x \in A n i = false. Proof. move=> fx0 i0; apply/negbTE; rewrite notin_set /A ltn_ord /= => -[_]. rewrite inE /= => -[r /=]; rewrite in_itv /= => /andP[r1 r2]. rewrite fx0 => -[r0]; move: r1 r2; rewrite {}r0 {r} => + r2. by rewrite ler_pdivr_mulr// mul0r lern0; exact/negP. Qed. Let fgen_A0 n x (i : 'I_(n * 2 ^ n)) : (n%:R%:E <= f x)%E -> x \in A n i = false. Proof. move=> fxn; apply/negbTE; rewrite /A ltn_ord. rewrite notin_set => /= -[_]; apply/negP. rewrite notin_set /= => -[r /=]. rewrite in_itv /= => /andP[r1 r2] rfx. move: fxn; rewrite -rfx lee_fin; apply/negP. rewrite -ltNge (lt_le_trans r2)// -natrX ler_pdivr_mulr//. by rewrite -natrM ler_nat (leq_trans (ltn_ord i)). Qed. Let disj_A0 x n (i k : 'I_(n * 2 ^ n)) : i != k -> x \in A n k -> (x \in A n i) = false. Proof. move=> ik xAn1k; apply/negbTE/negP => xAi. have /trivIsetP/(_ _ _ Logic.I Logic.I ik)/= := @trivIsetA n. rewrite predeqE => /(_ x)[+ _]. by rewrite 2!inE in xAn1k, xAi; move/(_ (conj xAi xAn1k)). Qed. Arguments disj_A0 {x n i} k. Let mB n : measurable (B n). Proof. exact: emeasurable_fun_c_infty. Qed. Let foo_B1 x n : D x -> f x = +oo%E -> x \in B n. Proof. by move=> Dx fxoo; rewrite /B inE /=; split => //=; rewrite /= fxoo leey. Qed. Let f0_B0 x n : f x = 0%:E -> n != 0%N -> (x \in B n) = false. Proof. move=> fx0 n0; apply/negP; rewrite inE /B /= => -[Dx] /=; apply/negP. by rewrite -ltNge fx0 lte_fin ltr0n lt0n. Qed. Let fgtn_B0 x n : (f x < n%:R%:E)%E -> (x \in B n) = false. Proof. move=> fxn; apply/negbTE/negP; rewrite inE /= => -[Dx] /=. by apply/negP; rewrite -ltNge. Qed. Let f0_approx0 n x : f x = 0%E -> approx n x = 0. Proof. move=> fx0; rewrite /approx; have [->|n0] := eqVneq n O. by rewrite mul0n mul0r addr0 big_ord0. rewrite f0_B0 // mulr0 addr0 big1 // => /= i _. have [->|i0] := eqVneq (nat_of_ord i) 0%N; first by rewrite mul0r mul0r. by rewrite f0_A0 // mulr0. Qed. Let fpos_approx_neq0 x : D x -> (0%E < f x < +oo)%E -> \forall n \near \oo, approx n x != 0. Proof. move=> Dx /andP[fx_gt0 fxoo]. have fxfin : f x \is a fin_num. by rewrite fin_numE; move: fxoo fx_gt0; case: (f x). rewrite -(fineK fxfin) lte_fin in fx_gt0. near=> n. rewrite /approx; apply/negP; rewrite paddr_eq0//; last exact: sumr_ge0. move/andP; rewrite psumr_eq0// => -[]/allP /= An0. rewrite mulf_eq0 => /orP[|]. by apply/negP; near: n; exists 1%N => //= m /=; rewrite lt0n pnatr_eq0. rewrite pnatr_eq0 => /eqP. have [//|] := boolP (x \in B n). rewrite notin_set /B /setI /= => /not_andP[] // /negP. rewrite -ltNge => fxn _. have K : (`|floor (fine (f x) * 2 ^+ n)| < n * 2 ^ n)%N. rewrite -ltz_nat gez0_abs; last by rewrite floor_ge0 mulr_ge0// ltW. rewrite -(@ltr_int R); rewrite (le_lt_trans (floor_le _)) // PoszM intrM. by rewrite -natrX ltr_pmul2r// -lte_fin (fineK fxfin). have xAnK : x \in A n (Ordinal K). rewrite inE /A /= K; split => //=. rewrite inE /=; exists (fine (f x)); last by rewrite fineK. rewrite in_itv /=; apply/andP; split. rewrite ler_pdivr_mulr// (le_trans _ (floor_le _))//. by rewrite -(@gez0_abs (floor _))// floor_ge0 mulr_ge0// ltW. rewrite ltr_pdivl_mulr// (lt_le_trans (lt_succ_Rfloor _))// RfloorE. rewrite -[in leRHS]addn1 natrD ler_add2r// -{1}(@gez0_abs (floor _))//. by rewrite floor_ge0// mulr_ge0// ltW. have /[!mem_index_enum]/(_ isT) := An0 (Ordinal K). apply/negP. rewrite xAnK mulr1 /= mulf_neq0// pnatr_eq0//= -lt0n absz_gt0 floor_neq0//. rewrite -ler_pdivr_mulr//; apply/orP; right; apply/ltW; near: n. exact: near_infty_natSinv_expn_lt (PosNum fx_gt0). Unshelve. all: by end_near. Qed. Let f_ub_approx n x : (f x < n%:R%:E)%E -> approx n x == 0 \/ exists k, [/\ (0 < k < n * 2 ^ n)%N, x \in A n k, approx n x = k%:R / 2 ^+ n & f x \in EFin @` [set` I n k]]. Proof. move=> fxn; rewrite /approx fgtn_B0 // mulr0 addr0. set lhs := (X in X == 0); have [|] := eqVneq lhs 0; first by left. rewrite {}/lhs psumr_eq0; last by move=> i _; rewrite mulr_ge0. move/allPn => [/= k _]. rewrite mulf_eq0 negb_or mulf_eq0 negb_or -andbA => /and3P[k_neq0 _]. rewrite pnatr_eq0 eqb0 negbK => xAnk. right. rewrite (bigD1 k) //= xAnk mulr1 big1 ?addr0; last first. by move=> i ik; rewrite (disj_A0 k)// mulr0. exists k; split => //. by rewrite lt0n ltn_ord andbT -(@pnatr_eq0 R). by move: xAnk; rewrite inE /A ltn_ord /= inE /= => -[/[swap] Dx]. Qed. Let notinD_A0 x n k : ~ D x -> (x \in A n k) = false. Proof. by move=> Dx; apply/negP; rewrite /A; case: ifPn => [?|_]; rewrite !inE => -[]. Qed. Let notinD_B0 x n : ~ D x -> (x \in B n) = false. Proof. by move=> Dx; apply/negP; rewrite inE => -[]. Qed. Lemma nd_approx : nondecreasing_seq approx. Proof. apply/nondecreasing_seqP => n; apply/lefP => x. have [Dx|Dx] := pselect (D x); last first. rewrite /approx big1; last by move=> i _; rewrite notinD_A0 // mulr0. rewrite notinD_B0// ?mulr0 addr0. rewrite big1; last by move=> i _; rewrite notinD_A0 // mulr0. by rewrite notinD_B0// ?mulr0 addr0. have [fxn|fxn] := ltP (f x) n%:R%:E. rewrite {2}/approx fgtn_B0 ?mulr0 ?addr0; last first. by rewrite (lt_trans fxn) // lte_fin ltr_nat. have [/eqP ->|[k [/andP[k0 kn] xAnk -> _]]] := f_ub_approx fxn. exact: sumr_ge0. move: (xAnk); rewrite inE {1}/A kn => -[_] /=. rewrite inE => -[r] /dyadic_itv_subU[|] rnk rfx. - have k2n : (k.*2 < n.+1 * 2 ^ n.+1)%N. rewrite expnS mulnCA mul2n ltn_double (ltn_trans kn) //. by rewrite ltn_mul2r expn_gt0 /= ltnS. rewrite (bigD1 (Ordinal k2n)) //=. have xAn1k : x \in A n.+1 k.*2. by rewrite inE /A k2n; split => //=; rewrite inE; exists r. rewrite xAn1k mulr1 big1 ?addr0; last first. by move=> i ik2n; rewrite (disj_A0 (Ordinal k2n)) ?mulr0. rewrite exprS invrM ?unitfE// -muln2 natrM -mulrA (mulrCA 2). by rewrite divrr ?mulr1 ?unitfE. - have k2n : (k.*2.+1 < n.+1 * 2 ^ n.+1)%N. move: kn; rewrite -ltn_double -(ltn_add2r 1) 2!addn1 => /leq_trans; apply. by rewrite -muln2 -mulnA -expnSr ltn_mul2r expn_gt0 /= ltnS. rewrite (bigD1 (Ordinal k2n)) //=. have xAn1k : x \in A n.+1 k.*2.+1. by rewrite /A /= k2n inE; split => //=; rewrite inE/=; exists r. rewrite xAn1k mulr1 big1 ?addr0; last first. by move=> i ik2n; rewrite (disj_A0 (Ordinal k2n)) // mulr0. rewrite -[leLHS]mulr1 -[X in _ * X <= _](@divrr _ 2%:R) ?unitfE//. rewrite mulf_div -natrM muln2 -natrX -natrM -expnSr natrX. by rewrite ler_pmul2r// ler_nat. have /orP[{}fxn|{}fxn] : ((n%:R%:E <= f x < n.+1%:R%:E) || (n.+1%:R%:E <= f x))%E. - by move: fxn; case: leP => /= [_ _|_ ->//]; rewrite orbT. - have [k [k1 k2]] := dyadic_itv_image fxn. have xBn : x \in B n. rewrite /B /= inE; split => //. by case/andP : fxn. rewrite /approx xBn mulr1 big1 ?add0r; last first. by move=> /= i _; rewrite fgen_A0 ?mulr0//; case/andP : fxn. rewrite fgtn_B0 ?mulr0 ?addr0; last by case/andP : fxn. have kn2 : (k < n.+1 * 2 ^ n.+1)%N by case/andP : k1 => _; rewrite mulnC. rewrite (bigD1 (Ordinal kn2)) //=. have xAn1k : x \in A n.+1 k by rewrite inE /A kn2. rewrite xAn1k mulr1 big1 ?addr0; last first. by move=> i /= ikn2; rewrite (disj_A0 (Ordinal kn2)) // mulr0. by rewrite -natrX ler_pdivl_mulr// mulrC -natrM ler_nat; case/andP : k1. - have xBn : x \in B n. by rewrite /B /= inE /= /= (le_trans _ fxn) // lee_fin ler_nat. rewrite /approx xBn mulr1. have xBn1 : x \in B n.+1 by rewrite /B /= inE. rewrite xBn1 mulr1 big1 ?add0r. by rewrite big1 ?add0r ?ler_nat// => /= i _; rewrite fgen_A0// mulr0. by move=> /= i _; rewrite fgen_A0 ?mulr0// (le_trans _ fxn)// lee_fin ler_nat. Qed. Lemma cvg_approx x (f0 : forall x, D x -> (0 <= f x)%E) : D x -> (f x < +oo)%E -> (approx^~ x) --> fine (f x). Proof. move=> Dx fxoo; have fxfin : f x \is a fin_num. rewrite fin_numE; apply/andP; split; last by rewrite lt_eqF. by rewrite gt_eqF // (lt_le_trans _ (f0 _ Dx)). apply/(@cvg_distP _ [normedModType R of R^o]) => _/posnumP[e]. rewrite near_map. have [fx0|fx0] := eqVneq (f x) 0%E. by near=> n; rewrite f0_approx0 // fx0 /= subrr normr0. have /(fpos_approx_neq0 Dx) [m _ Hm] : (0 < f x < +oo)%E. by rewrite fxoo andbT lt_neqAle eq_sym fx0 /= f0. near=> n. have mn : (m <= n)%N by near: n; exists m. have : fine (f x) < n%:R. near: n. exists `|floor (fine (f x))|.+1%N => //= p /=. rewrite -(@ler_nat R); apply: lt_le_trans. rewrite -addn1 natrD (_ : `| _ |%:R = (floor (fine (f x)))%:~R); last first. by rewrite -[in RHS](@gez0_abs (floor _))// floor_ge0 //; exact/le0R/f0. by rewrite -RfloorE lt_succ_Rfloor. rewrite -lte_fin (fineK fxfin) => fxn. have [approx_nx0|] := f_ub_approx fxn. by have := Hm _ mn; rewrite approx_nx0. move=> [k [/andP[k0 kn2n] ? ->]]. rewrite inE /= => -[r /=]. rewrite in_itv /= => /andP[k1 k2] rfx. rewrite (@le_lt_trans _ _ (1 / 2 ^+ n)) //. rewrite ler_norml; apply/andP; split. rewrite ler_subr_addl -mulrBl -lee_fin (fineK fxfin) -rfx lee_fin. rewrite (le_trans _ k1)// ler_pmul2r// -(@natrB _ _ 1) // ler_nat subn1. by rewrite leq_pred. rewrite ler_subl_addr -mulrDl -lee_fin -(natrD _ 1) add1n. by rewrite fineK// ltW// -rfx lte_fin. by near: n; exact: near_infty_natSinv_expn_lt. Unshelve. all: by end_near. Qed. Lemma le_approx k x (f0 : forall x, (0 <= f x)%E) : D x -> ((approx k x)%:E <= f x)%E. Proof. move=> Dx; have [fixoo|] := ltP (f x) (+oo%E); last first. by rewrite leye_eq => /eqP ->; rewrite leey. have nd_ag : {homo approx ^~ x : n m / (n <= m)%N >-> n <= m}. by move=> m n mn; exact/lefP/nd_approx. have fi0 y : D y -> (0 <= f y)%E by move=> ?; exact: f0. have cvg_af := cvg_approx fi0 Dx fixoo. have is_cvg_af : cvg (approx ^~ x) by apply/cvg_ex; eexists; exact: cvg_af. have {is_cvg_af} := nondecreasing_cvg_le nd_ag is_cvg_af k. rewrite -lee_fin => /le_trans; apply. rewrite -(@fineK _ (f x)); last first. by rewrite fin_numElt fixoo andbT (lt_le_trans _ (f0 _)). by move/(cvg_lim (@Rhausdorff R)) : cvg_af => ->. Qed. Lemma dvg_approx x : D x -> f x = +oo%E -> ~ cvg (approx^~ x : _ -> R^o). Proof. move=> Dx fxoo; have approx_x n : approx n x = n%:R. rewrite /approx foo_B1// mulr1 big1 ?add0r// => /= i _. by rewrite fgen_A0 // ?mulr0 // fxoo leey. case/cvg_ex => /= l; have [l0|l0] := leP 0%R l. - move=> /cvg_distP/(_ _ ltr01); rewrite near_map => -[n _]. move=> /(_ (`|ceil l|.+1 + n)%N) /= /(_ (leq_addl _ _)). rewrite approx_x. apply/negP; rewrite -leNgt distrC (le_trans _ (ler_sub_norm_add _ _)) //. rewrite normrN ler_subr_addl addSnnS [leRHS]ger0_norm ?ler0n//. rewrite natrD ler_add// ?ler1n// ger0_norm // (le_trans (ceil_ge _)) //. by rewrite -(@gez0_abs (ceil _)) // ceil_ge0. - move/cvg_distP => /(_ _ ltr01); rewrite near_map => -[n _]. move=> /(_ (`|floor l|.+1 + n)%N) /= /(_ (leq_addl _ _)). rewrite approx_x. apply/negP; rewrite -leNgt distrC (le_trans _ (ler_sub_norm_add _ _)) //. rewrite normrN ler_subr_addl addSnnS [leRHS]ger0_norm ?ler0n//. rewrite natrD ler_add// ?ler1n// ler0_norm //; last by rewrite ltW. rewrite (@le_trans _ _ (- floor l)%:~R) //. by rewrite mulrNz ler_oppl opprK floor_le. by rewrite -(@lez0_abs (floor _)) // floor_le0 // ltW. Qed. Lemma ecvg_approx (f0 : forall x, D x -> (0 <= f x)%E) x : D x -> EFin \o approx^~x --> f x. Proof. move=> Dx; have := leey (f x); rewrite le_eqVlt => /predU1P[|] fxoo. have dvg_approx := dvg_approx Dx fxoo. have : {homo approx ^~ x : n m / (n <= m)%N >-> n <= m}. by move=> m n mn; have := nd_approx mn => /lefP; exact. move/nondecreasing_dvg_lt => /(_ dvg_approx). by rewrite fxoo; exact/dvg_ereal_cvg. rewrite -(@fineK _ (f x)); first exact: (cvg_comp (cvg_approx f0 Dx fxoo)). by rewrite fin_numElt fxoo andbT (lt_le_trans _ (f0 _ _)). Qed. Let k2n_ge0 n (k : 'I_(n * 2 ^ n)) : 0 <= k%:R * 2 ^- n :> R. Proof. by []. Qed. Definition nnsfun_approx : {nnsfun T >-> R}^nat := fun n => locked (add_nnsfun (sum_nnsfun (fun k => match Bool.bool_dec (k < (n * 2 ^ n))%N true with | left h => scale_nnsfun (indic_nnsfun _ (mA n k)) (k2n_ge0 (Ordinal h)) | right _ => nnsfun0 end) (n * 2 ^ n)%N) (scale_nnsfun (indic_nnsfun _ (mB n)) (ler0n _ n))). Lemma nnsfun_approxE n : nnsfun_approx n = approx n :> (T -> R). Proof. rewrite funeqE => t /=; rewrite /nnsfun_approx; unlock; rewrite /=. rewrite sum_nnsfunE; congr (_ + _). by apply: eq_bigr => i _; case: Bool.bool_dec => [h|/negP]; [|rewrite ltn_ord]. Qed. Lemma cvg_nnsfun_approx (f0 : forall x, D x -> (0 <= f x)%E) x : D x -> EFin \o nnsfun_approx^~x --> f x. Proof. by move=> Dx; under eq_fun do rewrite nnsfun_approxE; exact: ecvg_approx. Qed. Lemma nd_nnsfun_approx : nondecreasing_seq (nnsfun_approx : (T -> R)^nat). Proof. move=> m n mn; rewrite (nnsfun_approxE n) (nnsfun_approxE m). exact: nd_approx. Qed. Lemma approximation : (forall t, D t -> (0 <= f t)%E) -> exists g : {nnsfun T >-> R}^nat, nondecreasing_seq (g : (T -> R)^nat) /\ (forall x, D x -> EFin \o g^~x --> f x). Proof. exists nnsfun_approx; split; [exact: nd_nnsfun_approx|]. move=> x Dx; exact: cvg_nnsfun_approx. Qed. End approximation. Section semi_linearity0. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D) (f1 f2 : T -> \bar R). Hypothesis f10 : forall x, D x -> 0 <= f1 x. Hypothesis mf1 : measurable_fun D f1. Lemma ge0_integralM_EFin k : (0 <= k)%R -> \int[mu]_(x in D) (k%:E * f1 x) = k%:E * \int[mu]_(x in D) f1 x. Proof. rewrite integral_mkcond erestrict_scale [in RHS]integral_mkcond => k0. set h1 := f1 \_ D. have h10 x : 0 <= h1 x by apply: erestrict_ge0. have mh1 : measurable_fun setT h1 by apply/(measurable_restrict _ mD). have [g [nd_g gh1]] := approximation measurableT mh1 (fun x _ => h10 x). pose kg := fun n => scale_nnsfun (g n) k0. rewrite (@nd_ge0_integral_lim _ _ _ mu (fun x => k%:E * h1 x) kg). - rewrite (_ : _ \o _ = fun n => sintegral mu (scale_nnsfun (g n) k0))//. rewrite (_ : (fun _ => _) = (fun n => k%:E * sintegral mu (g n))). rewrite ereal_limrM //; last first. by apply: is_cvg_sintegral => // x m n mn; apply/(lef_at x nd_g). by rewrite -(nd_ge0_integral_lim mu h10) // => x; [exact/(lef_at x nd_g)|exact: gh1]. by under eq_fun do rewrite (sintegralrM mu k (g _)). - by move=> t; rewrite mule_ge0. - by move=> x m n mn; rewrite /kg ler_pmul//; exact/lefP/nd_g. - move=> x. rewrite [X in X --> _](_ : _ = (fun n => k%:E * (g n x)%:E)) ?funeqE//. by apply: ereal_cvgrM => //; exact: gh1. Qed. End semi_linearity0. Section semi_linearity. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D) (f1 f2 : T -> \bar R). Hypothesis f10 : forall x, D x -> 0 <= f1 x. Hypothesis mf1 : measurable_fun D f1. Hypothesis f20 : forall x, D x -> 0 <= f2 x. Hypothesis mf2 : measurable_fun D f2. Lemma ge0_integralD : \int[mu]_(x in D) (f1 x + f2 x) = \int[mu]_(x in D) f1 x + \int[mu]_(x in D) f2 x. Proof. rewrite !(integral_mkcond D) erestrictD. set h1 := f1 \_ D; set h2 := f2 \_ D. have h10 x : 0 <= h1 x by apply: erestrict_ge0. have h20 x : 0 <= h2 x by apply: erestrict_ge0. have mh1 : measurable_fun setT h1 by apply/(measurable_restrict _ mD). have mh2 : measurable_fun setT h2 by apply/(measurable_restrict _ mD). have [g1 [nd_g1 gh1]] := approximation measurableT mh1 (fun x _ => h10 x). have [g2 [nd_g2 gh2]] := approximation measurableT mh2 (fun x _ => h20 x). pose g12 := fun n => add_nnsfun (g1 n) (g2 n). rewrite (@nd_ge0_integral_lim _ _ _ mu _ g12) //; last 3 first. - by move=> x; rewrite adde_ge0. - by apply: nondecreasing_seqD => // x; [exact/(lef_at x nd_g1)|exact/(lef_at x nd_g2)]. - move=> x Dx. rewrite (_ : _ \o _ = (fun n => (g1 n x)%:E + (g2 n x)%:E)) ?funeqE//. apply: ereal_cvgD => //; [|exact: gh1|exact: gh2]. by apply: ge0_adde_def => //; rewrite !inE; [exact: h10|exact: h20]. rewrite (_ : _ \o _ = fun n => sintegral mu (g1 n) + sintegral mu (g2 n)); last first. by rewrite funeqE => n /=; rewrite sintegralD. rewrite (nd_ge0_integral_lim _ _ (fun x => lef_at x nd_g1)) //; last first. by move=> x; exact: gh1. rewrite (nd_ge0_integral_lim _ _ (fun x => lef_at x nd_g2)) //; last first. by move=> x; exact: gh2. rewrite ereal_limD //. by apply: is_cvg_sintegral => // x Dx; exact/(lef_at x nd_g1). by apply: is_cvg_sintegral => // x Dx; exact/(lef_at x nd_g2). rewrite ge0_adde_def => //; rewrite inE; apply: ereal_lim_ge. - by apply: is_cvg_sintegral => // x Dx; exact/(lef_at x nd_g1). - by apply: nearW => n; exact: sintegral_ge0. - by apply: is_cvg_sintegral => // x Dx; exact/(lef_at x nd_g2). - by apply: nearW => n; exact: sintegral_ge0. Qed. Lemma ge0_le_integral : (forall x, D x -> f1 x <= f2 x) -> \int[mu]_(x in D) f1 x <= \int[mu]_(x in D) f2 x. Proof. move=> f12; rewrite !(integral_mkcond D). set h1 := f1 \_ D; set h2 := f2 \_ D. have h10 x : 0 <= h1 x by apply: erestrict_ge0. have h20 x : 0 <= h2 x by apply: erestrict_ge0. have mh1 : measurable_fun setT h1 by apply/(measurable_restrict _ mD). have mh2 : measurable_fun setT h2 by apply/(measurable_restrict _ mD). have h12 x : h1 x <= h2 x by apply: lee_restrict. have [g1 [nd_g1 /(_ _ Logic.I)gh1]] := approximation measurableT mh1 (fun x _ => h10 _). rewrite (nd_ge0_integral_lim _ h10 (fun x => lef_at x nd_g1) gh1)//. apply: ereal_lim_le. by apply: is_cvg_sintegral => // t Dt; exact/(lef_at t nd_g1). near=> n; rewrite ge0_integralTE//; apply: ereal_sup_ub => /=. exists (g1 n) => // t; rewrite (le_trans _ (h12 _)) //. have := gh1 t. have := leey (h1 t); rewrite le_eqVlt => /predU1P[->|ftoo]. by rewrite leey. have h1tfin : h1 t \is a fin_num. by rewrite fin_numE gt_eqF/= ?lt_eqF// (lt_le_trans _ (h10 t)). have := gh1 t. rewrite -(fineK h1tfin) => /ereal_cvg_real[ft_near]. set u_ := (X in X --> _) => u_h1 g1h1. have <- : lim u_ = fine (h1 t) by apply/cvg_lim => //; exact: Rhausdorff. rewrite lee_fin; apply: nondecreasing_cvg_le. by move=> // a b ab; rewrite /u_ /=; exact/lefP/nd_g1. by apply/cvg_ex; eexists; exact: u_h1. Unshelve. all: by end_near. Qed. End semi_linearity. Lemma emeasurable_funN d (T : measurableType d) (R : realType) D (f : T -> \bar R) : measurable D -> measurable_fun D f -> measurable_fun D (fun x => - f x)%E. Proof. by move=> mD mf; apply: measurable_fun_comp => //; exact: emeasurable_fun_minus. Qed. Section approximation_sfun. Variables (d : measure_display) (T : measurableType d) (R : realType) (f : T -> \bar R). Variables (D : set T) (mD : measurable D) (mf : measurable_fun D f). Lemma approximation_sfun : exists g : {sfun T >-> R}^nat, (forall x, D x -> EFin \o g^~x --> f x). Proof. have fp0 : (forall x, 0 <= f^\+ x)%E by []. have mfp : measurable_fun D f^\+%E. by apply: emeasurable_fun_max => //; exact: measurable_fun_cst. have fn0 : (forall x, 0 <= f^\- x)%E by []. have mfn : measurable_fun D f^\-%E. by apply: emeasurable_fun_max => //; [exact: emeasurable_funN | exact: measurable_fun_cst]. have [fp_ [fp_nd fp_cvg]] := approximation mD mfp (fun x _ => fp0 x). have [fn_ [fn_nd fn_cvg]] := approximation mD mfn (fun x _ => fn0 x). exists (fun n => [the {sfun T >-> R} of fp_ n \+ cst (-1) \* fn_ n]) => x /=. rewrite [X in X --> _](_ : _ = EFin \o fp_^~ x \+ (-%E \o EFin \o fn_^~ x))%E; last first. by apply/funext => n/=; rewrite EFinD mulN1r. by move=> Dx; rewrite (funeposneg f); apply: ereal_cvgD; [exact: add_def_funeposneg|apply: fp_cvg|apply:ereal_cvgN; exact: fn_cvg]. Qed. End approximation_sfun. Section emeasurable_fun. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). Implicit Types (D : set T) (f g : T -> \bar R). Lemma emeasurable_funD D f g : measurable_fun D f -> measurable_fun D g -> measurable_fun D (f \+ g). Proof. move=> mf mg mD. have Cnoom : measurable (~` [set -oo] : set (\bar R)) by apply: measurableC. have Cpoom : measurable (~` [set +oo] : set (\bar R)) by apply: measurableC. have mfg : measurable (D `&` [set x | f x +? g x]). suff -> : [set x | f x +? g x] = (f @^-1` (~` [set +oo]) `|` g @^-1` (~` [set -oo])) `&` (f @^-1` (~` [set -oo]) `|` g @^-1` (~` [set +oo])). by rewrite setIIr; apply: measurableI; rewrite setIUr; apply: measurableU; do ?[apply: mf|apply: mg]. apply/predeqP=> x; rewrite /preimage/= /adde_def !(negb_and, negb_or). by rewrite !(rwP2 eqP idP) !(rwP2 negP idP) !(rwP2 orP idP) !(rwP2 andP idP). wlog fg : D mD mf mg mfg / forall x, D x -> f x +? g x => [hwlogD|]; last first. have [f_ f_cvg] := approximation_sfun mD mf. have [g_ g_cvg] := approximation_sfun mD mg. apply: (emeasurable_fun_cvg (fun n x => (f_ n x + g_ n x)%:E)) => //. move=> n; apply/EFin_measurable_fun. by apply: (@measurable_funS _ _ _ _ setT) => //; exact: measurable_funD. move=> x Dx; under eq_fun do rewrite EFinD. by apply: ereal_cvgD; [exact: fg|exact: f_cvg|exact: g_cvg]. move=> A mA; wlog NAnoo: A mD mf mg mA / ~ (A -oo) => [hwlogA|]. have [] := pselect (A -oo); last exact: hwlogA. move=> /(@setD1K _ -oo)<-; rewrite preimage_setU setIUr. apply: measurableU; last by apply: hwlogA=> //; [exact: measurableD|case=>/=]. have -> : (f \+ g) @^-1` [set -oo] = f @^-1` [set -oo] `|` g @^-1` [set -oo]. apply/seteqP; split=> x /= => [/eqP|[]]; rewrite /preimage/=. - by rewrite adde_eq_ninfty => /orP[] /eqP->; [left|right]. - by move->. - by move->; rewrite addeC. by rewrite setIUr; apply: measurableU; [apply: mf|apply: mg]. have-> : D `&` (f \+ g) @^-1` A = (D `&` [set x | f x +? g x]) `&` (f \+ g) @^-1` A. rewrite -setIA; congr (_ `&` _). apply/seteqP; split=> x; rewrite /preimage/=; last by case. move=> Afgx; split=> //. by case: (f x) (g x) Afgx => [rf||] [rg||]. have Dfg : D `&` [set x | f x +? g x] `<=` D by apply: subIset; left. apply: hwlogD => //. - by apply: (measurable_funS mD) => //; do ?exact: measurableI. - by apply: (measurable_funS mD) => //; do ?exact: measurableI. - by rewrite -setIA setIid. - by move=> ? []. Qed. Lemma emeasurable_fun_sum D I s (h : I -> (T -> \bar R)) : (forall n, measurable_fun D (h n)) -> measurable_fun D (fun x => \sum_(i <- s) h i x). Proof. elim: s => [|s t ih] mf. by under eq_fun do rewrite big_nil; exact: measurable_fun_cst. under eq_fun do rewrite big_cons //=; apply: emeasurable_funD => //. exact: ih. Qed. Lemma ge0_emeasurable_fun_sum D (h : nat -> (T -> \bar R)) : (forall k x, 0 <= h k x) -> (forall k, measurable_fun D (h k)) -> measurable_fun D (fun x => \sum_(i h0 mh; rewrite [X in measurable_fun _ X](_ : _ = (fun x => elim_sup (fun n => \sum_(0 <= i < n) h i x))); last first. apply/funext=> x; rewrite is_cvg_elim_supE//. exact: is_cvg_ereal_nneg_natsum. by apply: measurable_fun_elim_sup => k; exact: emeasurable_fun_sum. Qed. Lemma emeasurable_funB D f g : measurable_fun D f -> measurable_fun D g -> measurable_fun D (f \- g). Proof. by move=> mf mg mD; apply: emeasurable_funD => //; exact: emeasurable_funN. Qed. Lemma emeasurable_funM D f g : measurable_fun D f -> measurable_fun D g -> measurable_fun D (f \* g). Proof. move=> mf mg mD. have m0 : measurable ([set 0] : set (\bar R)) by []. have mC0 : measurable ([set~ 0] : set (\bar R)) by apply: measurableC. have mCoo : measurable (~` [set -oo; +oo] : set (\bar R)). exact/measurableC/measurableU. have mfg : measurable (D `&` [set x | f x *? g x]). suff -> : [set x | f x *? g x] = (f @^-1` (~` [set 0]) `|` g @^-1` (~` [set -oo; +oo])) `&` (g @^-1` (~` [set 0]) `|` f @^-1` (~` [set -oo; +oo])). by rewrite setIIr; apply: measurableI; rewrite setIUr; apply: measurableU; do ?[apply: mf|apply: mg]. apply/predeqP=> x; rewrite /preimage/= /mule_def !(negb_and, negb_or). rewrite !(rwP2 eqP idP) !(rwP2 negP idP) !(rwP2 orP idP). rewrite !(rwP2 negP idP) !(rwP2 orP idP) !(rwP2 andP idP). rewrite eqe_absl leey andbT (orbC (g x == +oo)). by rewrite eqe_absl leey andbT (orbC (f x == +oo)). wlog fg : D mD mf mg mfg / forall x, D x -> f x *? g x => [hwlogM|]; last first. have [f_ f_cvg] := approximation_sfun mD mf. have [g_ g_cvg] := approximation_sfun mD mg. apply: (emeasurable_fun_cvg (fun n x => (f_ n x * g_ n x)%:E)) => //. move=> n; apply/EFin_measurable_fun. by apply: measurable_funM => //; exact: (@measurable_funS _ _ _ _ setT). move=> x Dx; under eq_fun do rewrite EFinM. by apply: ereal_cvgM; [exact: fg|exact: f_cvg|exact: g_cvg]. move=> A mA; wlog NA0: A mD mf mg mA / ~ (A 0) => [hwlogA|]. have [] := pselect (A 0); last exact: hwlogA. move=> /(@setD1K _ 0)<-; rewrite preimage_setU setIUr. apply: measurableU; last by apply: hwlogA=> //; [exact: measurableD|case => /=]. have -> : (fun x => f x * g x) @^-1` [set 0] = f @^-1` [set 0] `|` g @^-1` [set 0]. apply/seteqP; split=> x /= => [/eqP|[]]; rewrite /preimage/=. by rewrite mule_eq0 => /orP[] /eqP->; [left|right]. by move=> ->; rewrite mul0e. by move=> ->; rewrite mule0. by rewrite setIUr; apply: measurableU; [apply: mf|apply: mg]. have-> : D `&` (fun x => f x * g x) @^-1` A = (D `&` [set x | f x *? g x]) `&` (fun x => f x * g x) @^-1` A. rewrite -setIA; congr (_ `&` _). apply/seteqP; split=> x; rewrite /preimage/=; last by case. move=> Afgx; split=> //; apply: neq0_mule_def. by apply: contra_notT NA0; rewrite negbK => /eqP <-. have Dfg : D `&` [set x | f x *? g x] `<=` D by apply: subIset; left. apply: hwlogM => //. - by apply: (measurable_funS mD) => //; do ?exact: measurableI. - by apply: (measurable_funS mD) => //; do ?exact: measurableI. - by rewrite -setIA setIid. - by move=> ? []. Qed. Lemma emeasurable_funeM D (f : T -> \bar R) (k : \bar R) : measurable_fun D f -> measurable_fun D (fun x => k * f x)%E. Proof. move=> mf; rewrite (_ : (fun x => k * f x) = (cst k) \* f)//. exact/(emeasurable_funM _ mf)/measurable_fun_cst. Qed. End emeasurable_fun. Section ge0_integral_sum. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D) (I : Type) (f : I -> (T -> \bar R)). Hypothesis mf : forall n, measurable_fun D (f n). Hypothesis f0 : forall n x, D x -> 0 <= f n x. Lemma ge0_integral_sum (s : seq I) : \int[mu]_(x in D) (\sum_(k <- s) f k x) = \sum_(k <- s) \int[mu]_(x in D) (f k x). Proof. elim: s => [|h t ih]. by (under eq_fun do rewrite big_nil); rewrite big_nil integral0. rewrite big_cons /= -ih -ge0_integralD//. - by apply: eq_integral => x Dx; rewrite big_cons. - by move=> *; apply: f0. - by move=> *; apply: sume_ge0 => // k _; exact: f0. - exact: emeasurable_fun_sum. Qed. End ge0_integral_sum. Section monotone_convergence_theorem. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D) (g' : (T -> \bar R)^nat). Hypothesis mg' : forall n, measurable_fun D (g' n). Hypothesis g'0 : forall n x, D x -> 0 <= g' n x. Hypothesis nd_g' : forall x, D x -> nondecreasing_seq (g'^~ x). Let f' := fun x => lim (g'^~ x). Let g n := (g' n \_ D). Let g0 n x : 0 <= g n x. Proof. exact/erestrict_ge0/g'0. Qed. Let mg n : measurable_fun setT (g n). Proof. exact/(measurable_restrict _ mD). Qed. Let nd_g x : nondecreasing_seq (g^~ x). Proof. by move=> m n mn; rewrite /g/patch; case: ifP => // /set_mem /nd_g' ->. Qed. Let f := fun x => lim (g^~ x). Let is_cvg_g t : cvg (g^~ t). Proof. by move=> ?; apply: ereal_nondecreasing_is_cvg => m n ?; apply/nd_g. Qed. Local Definition g2' n : (T -> R)^nat := approx setT (g n). Local Definition g2 n : {nnsfun T >-> R}^nat := nnsfun_approx measurableT (mg n). Local Definition max_g2' : (T -> R)^nat := fun k t => (\big[maxr/0]_(i < k) (g2' i k) t)%R. Local Definition max_g2 : {nnsfun T >-> R}^nat := fun k => bigmax_nnsfun (g2^~ k) k. Let is_cvg_g2 n t : cvg (EFin \o (g2 n ^~ t)). Proof. apply: ereal_nondecreasing_is_cvg => a b ab. by rewrite lee_fin 2!nnsfun_approxE; exact/lefP/nd_approx. Qed. Let nd_max_g2 : nondecreasing_seq (max_g2 : (T -> R)^nat). Proof. apply/nondecreasing_seqP => n; apply/lefP => x; rewrite 2!bigmax_nnsfunE. rewrite (@le_trans _ _ (\big[maxr/0]_(i < n) g2 i n.+1 x)%R) //. apply: le_bigmax => i _; apply: (nondecreasing_seqP (g2 i ^~ x)).2 => a b ab. by rewrite !nnsfun_approxE; exact/lefP/nd_approx. rewrite (bigmaxD1 ord_max)// le_maxr; apply/orP; right. rewrite [leRHS](eq_bigl (fun i => nat_of_ord i < n)%N); last first. move=> i /=; rewrite neq_lt; apply/orP/idP => [[//|]|]; last by left. by move=> /(leq_trans (ltn_ord i)); rewrite ltnn. by rewrite (big_ord_narrow (leqnSn n)). Qed. Let is_cvg_max_g2 t : cvg (EFin \o max_g2 ^~ t). Proof. apply: ereal_nondecreasing_is_cvg => m n mn; rewrite lee_fin. exact/lefP/nd_max_g2. Qed. Let max_g2_g k x : ((max_g2 k x)%:E <= g k x)%O. Proof. rewrite bigmax_nnsfunE. apply: (@le_trans _ _ (\big[maxe/0%:E]_(i < k) g k x)); last first. apply/bigmax_lerP; split => //; apply: g0D. rewrite (@big_morph _ _ EFin 0%:E maxe) //; last by move=> *; rewrite maxEFin. apply: le_bigmax => i _; rewrite nnsfun_approxE /=. by rewrite (le_trans (le_approx _ _ _)) => //; exact/nd_g/ltnW. Qed. Let lim_max_g2_f t : lim (EFin \o max_g2 ^~ t) <= f t. Proof. by apply: lee_lim => //; near=> n; exact/max_g2_g. Unshelve. all: by end_near. Qed. Let lim_g2_max_g2 t n : lim (EFin\o g2 n ^~ t) <= lim (EFin \o max_g2 ^~ t). Proof. apply: lee_lim => //; near=> k; rewrite /= bigmax_nnsfunE lee_fin. have nk : (n < k)%N by near: k; exists n.+1. exact: (@bigmax_sup _ _ _ (Ordinal nk)). Unshelve. all: by end_near. Qed. Let cvg_max_g2_f t : EFin \o max_g2 ^~ t --> f t. Proof. have /cvg_ex[l g_l] := @is_cvg_max_g2 t. suff : l == f t by move=> /eqP <-. rewrite eq_le; apply/andP; split. by rewrite /f (le_trans _ (lim_max_g2_f _)) // (cvg_lim _ g_l). have := leey l; rewrite le_eqVlt => /predU1P[->|loo]; first by rewrite leey. rewrite -(cvg_lim _ g_l) //= ereal_lim_le => //. near=> n. have := leey (g n t); rewrite le_eqVlt => /predU1P[|] fntoo. have h := @dvg_approx _ _ _ setT _ t Logic.I fntoo. have g2oo : lim (EFin \o g2 n ^~ t) = +oo. apply/cvg_lim => //; apply/dvg_ereal_cvg. under [X in X --> _]eq_fun do rewrite nnsfun_approxE. have : {homo (approx setT (g n))^~ t : n0 m / (n0 <= m)%N >-> (n0 <= m)%R}. exact/lef_at/nd_approx. by move/nondecreasing_dvg_lt => /(_ h). have -> : lim (EFin \o max_g2 ^~ t) = +oo. by have := lim_g2_max_g2 t n; rewrite g2oo leye_eq => /eqP. by rewrite leey. - have approx_g_g := @cvg_approx _ _ _ setT _ t (fun t _ => g0 n t) Logic.I fntoo. have <- : lim (EFin \o g2 n ^~ t) = g n t. have /cvg_lim <- // : EFin \o (approx setT (g n)) ^~ t --> g n t. move/cvg_comp : approx_g_g; apply. by rewrite -(@fineK _ (g n t))// ge0_fin_numE// g0. rewrite (_ : _ \o _ = EFin \o approx setT (g n) ^~ t)// funeqE => m. by rewrite [in RHS]/= -nnsfun_approxE. exact: (le_trans _ (lim_g2_max_g2 t n)). Unshelve. all: by end_near. Qed. Lemma monotone_convergence : \int[mu]_(x in D) (f' x) = lim (fun n => \int[mu]_(x in D) (g' n x)). Proof. rewrite integral_mkcond. under [in RHS]eq_fun do rewrite integral_mkcond -/(g _). have -> : f' \_ D = f. apply/funext => x; rewrite /f /f' /g /patch /=; case: ifPn => //=. by rewrite lim_cst. apply/eqP; rewrite eq_le; apply/andP; split; last first. have nd_int_g : nondecreasing_seq (fun n => \int[mu]_x g n x). move=> m n mn; apply: ge0_le_integral => //. by move=> *; exact: nd_g. have ub n : \int[mu]_x g n x <= \int[mu]_x f x. apply: ge0_le_integral => //. - by move=> x _; apply: ereal_lim_ge => //; apply: nearW => k; exact/g0. - apply: emeasurable_fun_cvg mg _ => x _. exact: ereal_nondecreasing_is_cvg. - move=> x Dx; apply: ereal_lim_ge => //. near=> m; have nm : (n <= m)%N by near: m; exists n. exact/nd_g. by apply: ereal_lim_le => //; [exact:ereal_nondecreasing_is_cvg|exact:nearW]. rewrite (@nd_ge0_integral_lim _ _ _ mu _ max_g2) //; last 2 first. - by move=> t; apply: ereal_lim_ge => //; apply: nearW => n; exact: g0. - by move=> t m n mn; exact/lefP/nd_max_g2. apply: lee_lim. - by apply: is_cvg_sintegral => // t m n mn; exact/lefP/nd_max_g2. - apply: ereal_nondecreasing_is_cvg => // n m nm; apply: ge0_le_integral => //. by move=> *; exact/nd_g. - apply: nearW => n; rewrite ge0_integralTE//. by apply: ereal_sup_ub; exists (max_g2 n) => // t; exact: max_g2_g. Unshelve. all: by end_near. Qed. Lemma cvg_monotone_convergence : (fun n => \int[mu]_(x in D) g' n x) --> \int[mu]_(x in D) f' x. Proof. rewrite monotone_convergence; apply: ereal_nondecreasing_is_cvg => m n mn. by apply: ge0_le_integral => // t Dt; [exact: g'0|exact: g'0|exact: nd_g']. Qed. End monotone_convergence_theorem. Section integral_series. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). Variables (mu : {measure set T -> \bar R}) (D : set T) (mD : measurable D). Variable f : (T -> \bar R)^nat. Hypothesis mf : forall n, measurable_fun D (f n). Hypothesis f0 : forall n x, D x -> 0 <= f n x. Lemma integral_sum : \int[mu]_(x in D) (\sum_(n _) = (fun n => (\sum_(k < n) \int[mu]_(x in D) f k x)))//. by rewrite funeqE => n; rewrite ge0_integral_sum// big_mkord. - by move=> n; exact: emeasurable_fun_sum. - by move=> n x Dx; apply: sume_ge0 => m _; exact: f0. - by move=> x Dx m n mn; apply: lee_sum_nneg_natr => // k _ _; exact: f0. Qed. End integral_series. (* generalization of ge0_integralM_EFin to a constant potentially +oo using the monotone convergence theorem *) Section ge0_integralM. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D) (f : T -> \bar R). Hypothesis mf : measurable_fun D f. Lemma ge0_integralM (k : \bar R) : (forall x, D x -> 0 <= f x) -> 0 <= k -> \int[mu]_(x in D) (k * f x)%E = k * \int[mu]_(x in D) (f x). Proof. move=> f0; move: k => [k|_|//]; first exact: ge0_integralM_EFin. pose g : (T -> \bar R)^nat := fun n x => n%:R%:E * f x. have mg n : measurable_fun D (g n) by apply: emeasurable_funeM. have g0 n x : D x -> 0 <= g n x. by move=> Dx; apply: mule_ge0; [rewrite lee_fin|exact:f0]. have nd_g x : D x -> nondecreasing_seq (g^~x). by move=> Dx m n mn; rewrite lee_wpmul2r ?f0// lee_fin ler_nat. pose h := fun x => lim (g^~ x). transitivity (\int[mu]_(x in D) lim (g^~ x)). apply: eq_integral => x Dx; apply/esym/cvg_lim => //. have [fx0|fx0|fx0] := ltgtP 0 (f x). - rewrite gt0_mulye//; apply/ereal_cvgPpinfty => M M0. rewrite /g; case: (f x) fx0 => [r r0|_|//]; last first. exists 1%N => // m /= m0. by rewrite mulry gtr0_sg// ?mul1e ?leey// ltr0n. near=> n; rewrite lee_fin -ler_pdivr_mulr//. near: n; exists `|ceil (M / r)|%N => // m /=. rewrite -(ler_nat R); apply: le_trans. by rewrite natr_absz ger0_norm ?ceil_ge// ceil_ge0// divr_ge0// ltW. - rewrite lt0_mulye//; apply/ereal_cvgPninfty => M M0. rewrite /g; case: (f x) fx0 => [r r0|//|_]; last first. exists 1%N => // m /= m0. by rewrite mulrNy gtr0_sg// ?ltr0n// mul1e ?leNye. near=> n; rewrite lee_fin -ler_ndivr_mulr//. near: n; exists `|ceil (M / r)|%N => // m /=. rewrite -(ler_nat R); apply: le_trans. rewrite natr_absz ger0_norm ?ceil_ge// ceil_ge0// -mulrNN. by rewrite mulr_ge0// ler_oppr oppr0 ltW// invr_lt0. - rewrite -fx0 mule0 /g -fx0 [X in X @ _ --> _](_ : _ = cst 0). exact: cvg_cst. by rewrite funeqE => n /=; rewrite mule0. rewrite (monotone_convergence mu mD mg g0 nd_g). rewrite (_ : (fun _ => _) = (fun n => n%:R%:E * \int[mu]_(x in D) (f x))); last first. by rewrite funeqE => n; exact: ge0_integralM_EFin. have : 0 <= \int[mu]_(x in D) (f x) by exact: integral_ge0. rewrite le_eqVlt => /predU1P[<-|if_gt0]. rewrite mule0 (_ : (fun _ => _) = cst 0) ?lim_cst//. by under eq_fun do rewrite mule0. rewrite gt0_mulye//; apply/cvg_lim => //; apply/ereal_cvgPpinfty => M M0. near=> n; have [ifoo|] := ltP (\int[mu]_(x in D) (f x)) +oo; last first. rewrite leye_eq => /eqP ->; rewrite mulry muleC gt0_mulye ?leey//. by near: n; exists 1%N => // n /= n0; rewrite gtr0_sg// ?lte_fin// ltr0n. rewrite -(@fineK _ (\int[mu]_(x in D) f x)); last first. by rewrite fin_numElt ifoo andbT (le_lt_trans _ if_gt0). rewrite -lee_pdivr_mulr//; last first. by move: if_gt0 ifoo; case: (\int[mu]_(x in D) f x). near: n. exists `|ceil (M * (fine (\int[mu]_(x in D) f x))^-1)|%N => //. move=> n /=; rewrite -(@ler_nat R) -lee_fin; apply: le_trans. rewrite lee_fin natr_absz ger0_norm ?ceil_ge//. rewrite ceil_ge0// mulr_ge0 //; first exact: ltW. by rewrite invr_ge0; exact/le0R/integral_ge0. Unshelve. all: by end_near. Qed. End ge0_integralM. Section fatou. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D) (f : (T -> \bar R)^nat). Hypothesis mf : forall n, measurable_fun D (f n). Hypothesis f0 : forall n x, D x -> 0 <= f n x. Lemma fatou : \int[mu]_(x in D) elim_inf (f^~ x) <= elim_inf (fun n => \int[mu]_(x in D) f n x). Proof. pose g n := fun x => einfs (f ^~ x) n. have mg := measurable_fun_einfs mf. have g0 n x : D x -> 0 <= g n x. by move=> Dx; apply: lb_ereal_inf => _ [m /= nm <-]; exact: f0. rewrite monotone_convergence //; last first. move=> x Dx m n mn /=; apply: le_ereal_inf => _ /= [p /= np <-]. by exists p => //=; rewrite (leq_trans mn). apply: lee_lim. - apply/cvg_ex; eexists; apply/ereal_nondecreasing_cvg => a b ab. apply: ge0_le_integral => //; [exact: g0| exact: mg| exact: g0| exact: mg|]. move=> x Dx; apply: le_ereal_inf => _ [n /= bn <-]. by exists n => //=; rewrite (leq_trans ab). - apply/cvg_ex; eexists; apply/ereal_nondecreasing_cvg => a b ab. apply: le_ereal_inf => // _ [n /= bn <-]. by exists n => //=; rewrite (leq_trans ab). - apply: nearW => m. have : forall n p, (p >= n)%N -> \int[mu]_(x in D) g n x <= einfs (fun k => \int[mu]_(x in D) f k x) n. move=> n p np; apply: lb_ereal_inf => /= _ [k /= nk <-]. apply: ge0_le_integral => //; [exact: g0|exact: mg|exact: f0|]. by move=> x Dx; apply: ereal_inf_lb; exists k. exact. Qed. End fatou. Section integralN. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Lemma integralN D (f : T -> \bar R) : \int[mu]_(x in D) f^\+ x +? (- \int[mu]_(x in D) f^\- x) -> \int[mu]_(x in D) - f x = - \int[mu]_(x in D) f x. Proof. have [f_fin _|] := boolP (\int[mu]_(x in D) f^\- x \is a fin_num). rewrite integralE// [in RHS]integralE// oppeD ?fin_numN// oppeK addeC. by rewrite funenegN. rewrite fin_numE negb_and 2!negbK => /orP[nfoo|/eqP nfoo]. exfalso; move/negP : nfoo; apply; rewrite -leeNy_eq; apply/negP. by rewrite -ltNge (lt_le_trans _ (integral_ge0 _ _)). rewrite nfoo adde_defEninfty. rewrite -leye_eq -ltNge lte_pinfty_eq => /orP[f_fin|/eqP pfoo]. rewrite integralE// [in RHS]integralE// nfoo [in RHS]addeC oppeD//. by rewrite funenegN. by rewrite integralE// [in RHS]integralE// funeposN funenegN nfoo pfoo. Qed. Lemma integral_ge0N (D : set T) (f : T -> \bar R) : (forall x, D x -> 0 <= f x) -> \int[mu]_(x in D) - f x = - \int[mu]_(x in D) f x. Proof. move=> f0; rewrite integralN // (eq_integral _ _ (ge0_funenegE _))// integral0. by rewrite oppe0 fin_num_adde_def. Qed. End integralN. Section integral_cst. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (f : T -> \bar R) (D : set T) (mD : measurable D). Lemma sintegral_cst (x : {nonneg R}) : sintegral mu (cst x%:num \_ D) = x%:num%:E * mu D. Proof. rewrite sintegralE (fsbig_widen _ [set 0%R; x%:num])/=; last 2 first. - by move=> y [t _ <-] /=; rewrite /patch; case: ifPn; [right|left]. - by move=> y [_ /=/preimage10->]; rewrite measure0 mule0. have [->|x0] := eqVneq x%:num 0%R; first by rewrite setUid fsbig_set1 !mul0e. rewrite fsbigU0//=; last by move=> y [->]/esym; apply/eqP. rewrite !fsbig_set1 mul0e add0e preimage_restrict//. by rewrite ifN ?set0U ?setIidl//= notin_set; apply/eqP; rewrite eq_sym. Qed. Lemma integral_cst (r : R) : \int[mu]_(x in D) (EFin \o cst r) x = r%:E * mu D. Proof. wlog r0 : r / (0 <= r)%R. move=> h; have [|r0] := leP 0%R r; first exact: h. rewrite -[in RHS](opprK r) EFinN mulNe -h ?oppr_ge0; last exact: ltW. rewrite -integral_ge0N//; last by move=> t ?; rewrite /= lee_fin oppr_ge0 ltW. by under [RHS]eq_integral do rewrite /= opprK. rewrite (eq_integral (EFin \o cst_nnsfun T (NngNum r0)))//. by rewrite integral_nnsfun// sintegral_cst. Qed. Lemma integral_cst_pinfty : mu D != 0 -> \int[mu]_(x in D) (cst +oo) x = +oo. Proof. move=> muD0; pose g : (T -> \bar R)^nat := fun n => cst n%:R%:E. have <- : (fun t => lim (g^~ t)) = cst +oo. rewrite funeqE => t; apply/cvg_lim => //=. apply/dvg_ereal_cvg/cvgPpinfty => M; exists `|ceil M|%N => //= m. rewrite /= -(ler_nat R); apply: le_trans. by rewrite (le_trans (ceil_ge _))// natr_absz ler_int ler_norm. rewrite monotone_convergence //. - rewrite /g (_ : (fun _ => _) = (fun n => n%:R%:E * mu D)); last first. by rewrite funeqE => n; rewrite -integral_cst. apply/cvg_lim => //; apply/ereal_cvgPpinfty => M M0. have [muDoo|muDoo] := ltP (mu D) +oo; last first. exists 1%N => // m /= m0; move: muDoo; rewrite leye_eq => /eqP ->. by rewrite mulry gtr0_sg ?mul1e ?leey// ltr0n. exists `|ceil (M / fine (mu D))|%N => // m /=. rewrite -(ler_nat R) => MDm. rewrite -(@fineK _ (mu D)); last by rewrite ge0_fin_numE//. rewrite -lee_pdivr_mulr; last first. by apply: lt0R; rewrite muDoo andbT lt_neqAle measure_ge0// andbT eq_sym. rewrite lee_fin; apply: le_trans MDm. by rewrite natr_absz (le_trans (ceil_ge _))// ler_int ler_norm. - by move=> n; exact: measurable_fun_cst. - by move=> n x Dx; rewrite lee_fin. - by move=> t Dt n m nm; rewrite /g lee_fin ler_nat. Qed. End integral_cst. Section integral_ind. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D). Lemma integral_indic (E : set T) : measurable E -> \int[mu]_(x in D) (\1_E x)%:E = mu (E `&` D). Proof. move=> mE; rewrite (_ : \1_E = indic_nnsfun R mE)// integral_nnsfun//=. by rewrite restrict_indic sintegral_indic//; exact: measurableI. Qed. End integral_ind. Section integralM_indic. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). Variables (m : {measure set T -> \bar R}) (D : set T) (mD : measurable D). Lemma integralM_indic (f : R -> set T) (k : R) : ((k < 0)%R -> f k = set0) -> measurable (f k) -> \int[m]_(x in D) (k * \1_(f k) x)%:E = k%:E * \int[m]_(x in D) (\1_(f k) x)%:E. Proof. move=> fk0 mfk; have [k0|k0] := ltP k 0%R. rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0; last first. by move=> x _; rewrite fk0// indic0 mulr0. rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0// => x _. by rewrite fk0// indic0. under eq_integral do rewrite EFinM. rewrite ge0_integralM//. - apply/EFin_measurable_fun/(@measurable_funS _ _ _ _ setT) => //. by rewrite (_ : \1_(f k) = mindic R mfk). - by move=> y _; rewrite lee_fin. Qed. Lemma integralM_indic_nnsfun (f : {nnsfun T >-> R}) (k : R) : \int[m]_(x in D) (k * \1_(f @^-1` [set k]) x)%:E = k%:E * \int[m]_(x in D) (\1_(f @^-1` [set k]) x)%:E. Proof. rewrite (@integralM_indic (fun k => f @^-1` [set k]))// => k0. by rewrite preimage_nnfun0. Qed. End integralM_indic. Section integral_dirac. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (a : T) (R : realType). Variables (D : set T) (mD : measurable D). Let ge0_integral_dirac (f : T -> \bar R) (mf : measurable_fun D f) (f0 : forall x, D x -> 0 <= f x) : D a -> \int[\d_a]_(x in D) (f x) = f a. Proof. move=> Da; have [f_ [ndf_ f_f]] := approximation mD mf f0. transitivity (lim (fun n => \int[\d_ a]_(x in D) (f_ n x)%:E)). rewrite -monotone_convergence//. - apply: eq_integral => x Dx; apply/esym/cvg_lim => //; apply: f_f. by rewrite inE in Dx. - by move=> n; apply/EFin_measurable_fun; exact/(@measurable_funS _ _ _ _ setT). - by move=> *; rewrite lee_fin. - by move=> x _ m n mn; rewrite lee_fin; exact/lefP/ndf_. rewrite (_ : (fun _ => _) = (fun n => (f_ n a)%:E)). by apply/cvg_lim => //; exact: f_f. apply/funext => n; under eq_integral do rewrite fimfunE -sumEFin. rewrite ge0_integral_sum//. - under eq_bigr; first by move=> r _; rewrite integralM_indic_nnsfun//; over. rewrite /= (big_fsetD1 (f_ n a)); last first. by rewrite in_fset_set// in_setE; exists a. rewrite integral_indic//= diracE mem_set// mule1. rewrite big1_fset ?adde0// => r; rewrite !inE => /andP[rfna _] _. rewrite integral_indic//= diracE memNset ?mule0//. by apply/not_andP; left; exact/nesym/eqP. - move=> r; apply/EFin_measurable_fun. apply: measurable_funM => //; first exact: measurable_fun_cst. apply: (@measurable_funS _ _ _ _ setT) => //. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP (f_ n) r)). - by move=> r x _; rewrite muleindic_ge0. Qed. Lemma integral_dirac (f : T -> \bar R) (mf : measurable_fun D f) : \int[\d_ a]_(x in D) f x = (\1_D a)%:E * f a. Proof. have [/[!inE] aD|aD] := boolP (a \in D). rewrite integralE ge0_integral_dirac//; last exact/emeasurable_fun_funepos. rewrite ge0_integral_dirac//; last exact/emeasurable_fun_funeneg. by rewrite [in RHS](funeposneg f) indicE mem_set// mul1e. rewrite indicE (negbTE aD) mul0e -(integral_measure_zero D f)//. apply: eq_measure_integral => //= S mS DS; rewrite /dirac indicE memNset// => /DS. by rewrite notin_set in aD. Qed. End integral_dirac. Section integral_measure_sum_nnsfun. Local Open Scope ereal_scope. Variables (d : _) (T : measurableType d) (R : realType). Variables (m_ : {measure set T -> \bar R}^nat) (N : nat). Let m := msum m_ N. Let integral_measure_sum_indic (E D : set T) (mE : measurable E) (mD : measurable D) : \int[m]_(x in E) (\1_D x)%:E = \sum_(n < N) \int[m_ n]_(x in E) (\1_D x)%:E. Proof. rewrite integral_indic//= /msum/=; apply eq_bigr => i _. by rewrite integral_indic// setIT. Qed. Let integralT_measure_sum (f : {nnsfun T >-> R}) : \int[m]_x (f x)%:E = \sum_(n < N) \int[m_ n]_x (f x)%:E. Proof. under eq_integral do rewrite fimfunE -sumEFin. rewrite ge0_integral_sum//; last 2 first. - move=> r /=; apply: measurable_fun_comp => //. apply: measurable_funM => //. exact: measurable_fun_cst. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f r)). - by move=> r t _; rewrite EFinM muleindic_ge0. transitivity (\sum_(i <- fset_set (range f)) (\sum_(n < N) i%:E * \int[m_ n]_x (\1_(f @^-1` [set i]) x)%:E)). apply eq_bigr => r _. rewrite integralM_indic_nnsfun// integral_measure_sum_indic//. by rewrite ge0_sume_distrr// => n _; apply integral_ge0 => t _; rewrite lee_fin. rewrite exchange_big/=; apply eq_bigr => i _. rewrite integralT_nnsfun sintegralE fsbig_finite//=; apply eq_bigr => r _. by congr (_ * _); rewrite integral_indic// setIT. Qed. Lemma integral_measure_sum_nnsfun (D : set T) (mD : measurable D) (f : {nnsfun T >-> R}) : \int[m]_(x in D) (f x)%:E = \sum_(n < N) \int[m_ n]_(x in D) (f x)%:E. Proof. rewrite integral_mkcond. transitivity (\int[m]_x (proj_nnsfun f mD x)%:E). by apply: eq_integral => t _ /=; rewrite /patch mindicE; case: ifPn => // tD; rewrite ?mulr1 ?mulr0. rewrite integralT_measure_sum; apply eq_bigr => i _. rewrite [RHS]integral_mkcond; apply: eq_integral => t _. rewrite /= /patch /mindic indicE. by case: (boolP (t \in D)) => tD; rewrite ?mulr1 ?mulr0. Qed. End integral_measure_sum_nnsfun. Lemma integral_measure_add_nnsfun (d : _) (T : measurableType d) (R : realType) (m1 m2 : {measure set T -> \bar R}) (D : set T) (mD : measurable D) (f : {nnsfun T >-> R}) : (\int[measure_add m1 m2]_(x in D) (f x)%:E = \int[m1]_(x in D) (f x)%:E + \int[m2]_(x in D) (f x)%:E)%E. Proof. rewrite /measureD integral_measure_sum_nnsfun// 2!big_ord_recl/= big_ord0. by rewrite adde0. Qed. Section integral_mfun_measure_sum. Local Open Scope ereal_scope. Variables (d : _) (T : measurableType d) (R : realType). Variable m_ : {measure set T -> \bar R}^nat. Lemma ge0_integral_measure_sum (D : set T) (mD : measurable D) (f : T -> \bar R) : (forall x, D x -> 0 <= f x) -> measurable_fun D f -> forall N, \int[msum m_ N]_(x in D) f x = \sum_(n < N) \int[m_ n]_(x in D) f x. Proof. move=> f0 mf. have [f_ [f_nd f_f]] := approximation mD mf f0. elim => [|N ih]; first by rewrite big_ord0 msum_mzero integral_measure_zero. rewrite big_ord_recr/= -ih. rewrite (_ : _ m_ N.+1 = measure_add [the measure _ _ of msum m_ N] (m_ N)); last first. by apply/funext => A; rewrite measure_addE /msum/= big_ord_recr. have mf_ n : measurable_fun D (fun x => (f_ n x)%:E). by apply: (@measurable_funS _ _ _ _ setT) => //; exact/EFin_measurable_fun. have f_ge0 n x : D x -> 0 <= (f_ n x)%:E by move=> Dx; rewrite lee_fin. have cvg_f_ (m : {measure set T -> \bar R}) : cvg (fun x => \int[m]_(x0 in D) (f_ x x0)%:E). apply: ereal_nondecreasing_is_cvg => a b ab. apply ge0_le_integral => //; [exact: f_ge0|exact: f_ge0|]. by move=> t Dt; rewrite lee_fin; apply/lefP/f_nd. transitivity (lim (fun n => \int[measure_add [the measure _ _ of msum m_ N] (m_ N)]_(x in D) (f_ n x)%:E)). rewrite -monotone_convergence//; last first. by move=> t Dt a b ab; rewrite lee_fin; exact/lefP/f_nd. by apply eq_integral => t /[!inE] Dt; apply/esym/cvg_lim => //; exact: f_f. transitivity (lim (fun n => \int[msum m_ N]_(x in D) (f_ n x)%:E + \int[m_ N]_(x in D) (f_ n x)%:E)). by congr (lim _); apply/funext => n; by rewrite integral_measure_add_nnsfun. rewrite ereal_limD//; last first. by apply: ge0_adde_def; rewrite inE; apply: ereal_lim_ge => //; apply: nearW => n; apply: integral_ge0 => //; exact: f_ge0. by congr (_ + _); (rewrite -monotone_convergence//; [ apply eq_integral => t /[!inE] Dt; apply/cvg_lim => //; exact: f_f | move=> t Dt a b ab; rewrite lee_fin; exact/lefP/f_nd]). Qed. End integral_mfun_measure_sum. Lemma integral_measure_add (d : _) (T : measurableType d) (R : realType) (m1 m2 : {measure set T -> \bar R}) (D : set T) (mD : measurable D) (f : T -> \bar R) : (forall x, D x -> 0 <= f x)%E -> measurable_fun D f -> (\int[measure_add m1 m2]_(x in D) f x = \int[m1]_(x in D) f x + \int[m2]_(x in D) f x)%E. Proof. move=> f0 mf; rewrite /measureD ge0_integral_measure_sum// 2!big_ord_recl/=. by rewrite big_ord0 adde0. Qed. Section integral_measure_series. Local Open Scope ereal_scope. Variables (d : _) (T : measurableType d) (R : realType). Variable m_ : {measure set T -> \bar R}^nat. Let m := mseries m_ O. Let integral_measure_series_indic (D : set T) (mD : measurable D) : \int[m]_x (\1_D x)%:E = \sum_(n i _. by rewrite integral_indic// setIT. Qed. Lemma integral_measure_series_nnsfun (D : set T) (mD : measurable D) (f : {nnsfun T >-> R}) : \int[m]_x (f x)%:E = \sum_(n r /=. apply: measurable_fun_comp => //. apply: measurable_funM => //; first exact: measurable_fun_cst. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f r)). - by move=> r t _; rewrite EFinM muleindic_ge0. transitivity (\sum_(i <- fset_set (range f)) (\sum_(n r _. rewrite integralM_indic_nnsfun// integral_measure_series_indic// nneseriesrM//. by move=> n _; apply integral_ge0 => t _; rewrite lee_fin. rewrite -nneseries_sum; last first. move=> r j _. have [r0|r0] := leP 0%R r. by rewrite mule_ge0//; apply integral_ge0 => // t _; rewrite lee_fin. rewrite (eq_integral (cst 0)) ?integral0 ?mule0// => t _. by rewrite preimage_nnfun0// indicE in_set0. apply eq_nneseries => k _. rewrite integralT_nnsfun sintegralE fsbig_finite//=; apply eq_bigr => r _. by congr (_ * _); rewrite integral_indic// setIT. Qed. End integral_measure_series. Section ge0_integral_measure_series. Local Open Scope ereal_scope. Variables (d : _) (T : measurableType d) (R : realType). Variable m_ : {measure set T -> \bar R}^nat. Let m := mseries m_ O. Lemma ge0_integral_measure_series (D : set T) (mD : measurable D) (f : T -> \bar R) : (forall t, D t -> 0 <= f t) -> measurable_fun D f -> \int[m]_(x in D) f x = \sum_(n f0 mf. apply/eqP; rewrite eq_le; apply/andP; split; last first. suff : forall n, \sum_(k < n) \int[m_ k]_(x in D) f x <= \int[m]_(x in D) f x. move=> n; apply: ereal_lim_le => //. by apply: is_cvg_ereal_nneg_natsum => k _; exact: integral_ge0. by apply: nearW => x; rewrite big_mkord. move=> n. rewrite [X in _ <= X](_ : _ = (\sum_(k < n) \int[m_ k]_(x in D) f x + \int[mseries m_ n]_(x in D) f x)); last first. transitivity (\int[measure_add [the measure _ _ of msum m_ n] [the measure _ _ of mseries m_ n]]_(x in D) f x). congr (\int[_]_(_ in D) _); apply/funext => A. by rewrite measure_addE; exact: nneseries_split. rewrite integral_measure_add//; congr (_ + _). by rewrite -ge0_integral_measure_sum. by apply: lee_addl; exact: integral_ge0. rewrite ge0_integralE//=; apply: ub_ereal_sup => /= _ [g /= gf] <-. rewrite -integralT_nnsfun (integral_measure_series_nnsfun _ mD). apply: lee_nneseries => n _. by apply integral_ge0 => // x _; rewrite lee_fin. rewrite [leRHS]integral_mkcond; apply ge0_le_integral => //. - by move=> x _; rewrite lee_fin. - exact/EFin_measurable_fun. - by move=> x _; rewrite erestrict_ge0. - exact/(measurable_restrict _ mD). Qed. End ge0_integral_measure_series. Section subset_integral. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Lemma integral_setU (A B : set T) (mA : measurable A) (mB : measurable B) (f : T -> \bar R) : measurable_fun (A `|` B) f -> (forall x, (A `|` B) x -> 0 <= f x) -> [disjoint A & B] -> \int[mu]_(x in A `|` B) f x = \int[mu]_(x in A) f x + \int[mu]_(x in B) f x. Proof. move=> mf f0 AB. transitivity (\int[mu]_(x in A `|` B) ((f \_ A) x + (f \_ B) x)). apply: eq_integral => x; rewrite inE => -[xA|xB]. rewrite /patch mem_set// ifF ?adde0//; apply/negbTE/negP; rewrite inE => xB. by move: AB; rewrite disj_set2E => /eqP; apply/eqP/set0P; exists x. rewrite /patch addeC mem_set// ifF ?adde0//; apply/negbTE/negP; rewrite inE => xA. by move: AB; rewrite disj_set2E => /eqP; apply/eqP/set0P; exists x. rewrite ge0_integralD//; last 5 first. - exact: measurableU. - by move=> x _; apply: erestrict_ge0 => y Ay; apply: f0; left. - have : measurable_fun A f. by apply: measurable_funS mf; [exact: measurableU|exact: subsetUl]. by apply/(measurable_restrict _ _ _ _).1 => //; exact: measurableU. - by move=> x _; apply: erestrict_ge0 => y By; apply: f0; right. - have : measurable_fun B f. by apply: measurable_funS mf; [exact: measurableU|exact: subsetUr]. by apply/(measurable_restrict _ _ _ _).1 => //; exact: measurableU. by rewrite -integral_mkcondl setIC setUK -integral_mkcondl setKU. Qed. Lemma subset_integral (A B : set T) (mA : measurable A) (mB : measurable B) (f : T -> \bar R) : measurable_fun B f -> (forall x, B x -> 0 <= f x) -> A `<=` B -> \int[mu]_(x in A) f x <= \int[mu]_(x in B) f x. Proof. move=> mf f0 AB; rewrite -(setDUK AB) integral_setU//; last 4 first. - exact: measurableD. - by rewrite setDUK. - by move=> x; rewrite setDUK//; exact: f0. - by rewrite disj_set2E setDIK. by apply: lee_addl; apply: integral_ge0 => x [Bx _]; exact: f0. Qed. Lemma integral_set0 (f : T -> \bar R) : \int[mu]_(x in set0) f x = 0. Proof. rewrite integral_mkcond (eq_integral (cst 0)) ?integral0// => x _. by rewrite /restrict; case: ifPn => //; rewrite in_set0. Qed. Lemma ge0_integral_bigsetU (F : (set T)^nat) (f : T -> \bar R) n : (forall n, measurable (F n)) -> let D := \big[setU/set0]_(i < n) F i in measurable_fun D f -> (forall x, D x -> 0 <= f x) -> trivIset `I_n F -> \int[mu]_(x in D) f x = \sum_(i < n) \int[mu]_(x in F i) f x. Proof. move=> mF. elim: n => [|n ih] D mf f0 tF; first by rewrite /D 2!big_ord0 integral_set0. rewrite /D big_ord_recr/= integral_setU//; last 4 first. - exact: bigsetU_measurable. - by move: mf; rewrite /D big_ord_recr. - by move: f0; rewrite /D big_ord_recr. - apply/eqP; move: (trivIset_bigsetUI tF (ltnSn n) (leqnn n)). rewrite [in X in X -> _](eq_bigl xpredT)// => i. by rewrite (leq_trans (ltn_ord i)). rewrite ih ?big_ord_recr//. - apply: measurable_funS mf => //; first exact: bigsetU_measurable. by rewrite /D big_ord_recr /=; apply: subsetUl. - by move=> t Dt; apply: f0; rewrite /D big_ord_recr/=; left. - by apply: sub_trivIset tF => x; exact: leq_trans. Qed. Lemma le_integral_abse (D : set T) (mD : measurable D) (g : T -> \bar R) a : measurable_fun D g -> (0 < a)%R -> a%:E * mu (D `&` [set x | (`|g x| >= a%:E)%E]) <= \int[mu]_(x in D) `|g x|. Proof. move=> mg a0; have ? : measurable (D `&` [set x | (a%:E <= `|g x|)%E]). by apply: emeasurable_fun_c_infty => //; exact: measurable_fun_comp. apply: (@le_trans _ _ (\int[mu]_(x in D `&` [set x | `|g x| >= a%:E]) `|g x|)). rewrite -integral_cst//; apply: ge0_le_integral => //. - by move=> x _ /=; rewrite ltW. - exact/EFin_measurable_fun/measurable_fun_cst. - by apply: measurable_fun_comp => //; exact: measurable_funS mg. - by move=> x /= []. by apply: subset_integral => //; exact: measurable_fun_comp. Qed. End subset_integral. Section Rintegral. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Definition Rintegral (D : set T) (f : T -> \bar R) := fine (\int[mu]_(x in D) f x). End Rintegral. Section integrable. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). Definition integrable (mu : set T -> \bar R) D f := measurable_fun D f /\ (\int[mu]_(x in D) `|f x| < +oo). Variables (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D). Implicit Type f g : T -> \bar R. Notation mu_int := (integrable mu D). Lemma integrable0 : mu_int (cst 0). Proof. split; first exact: measurable_fun_cst. under eq_integral do rewrite (gee0_abs (lexx 0)). by rewrite integral0. Qed. Lemma eq_integrable f g : {in D, f =1 g} -> mu_int f -> mu_int g. Proof. move=> fg [mf fi]; split; first exact: eq_measurable_fun mf. rewrite (le_lt_trans _ fi)//; apply: ge0_le_integral=> //. by apply: measurable_fun_comp => //; exact: eq_measurable_fun mf. by apply: measurable_fun_comp => //; exact: eq_measurable_fun mf. by move=> x Dx; rewrite fg// inE. Qed. Lemma le_integrable f g : measurable_fun D f -> (forall x, D x -> `|f x| <= `|g x|) -> mu_int g -> mu_int f. Proof. move=> mf fg [mfg goo]; split => //; rewrite (le_lt_trans _ goo) //. by apply: ge0_le_integral => //; exact: measurable_fun_comp. Qed. Lemma integrableN f : mu_int f -> mu_int (-%E \o f). Proof. move=> [mf foo]; split; last by rewrite /comp; under eq_fun do rewrite abseN. by rewrite /comp; apply: measurable_fun_comp =>//; exact: emeasurable_fun_minus. Qed. Lemma integrablerM (k : R) f : mu_int f -> mu_int (fun x => k%:E * f x). Proof. move=> [mf foo]; split; first exact: emeasurable_funeM. under eq_fun do rewrite abseM. by rewrite ge0_integralM// ?lte_mul_pinfty//; exact: measurable_fun_comp. Qed. Lemma integrableMr (k : R) f : mu_int f -> mu_int (f \* cst k%:E). Proof. by move=> mf; apply: eq_integrable (integrablerM k mf) => // x; rewrite muleC. Qed. Lemma integrableD f g : mu_int f -> mu_int g -> mu_int (f \+ g). Proof. move=> [mf foo] [mg goo]; split; first exact: emeasurable_funD. apply: (@le_lt_trans _ _ (\int[mu]_(x in D) (`|f x| + `|g x|))). apply: ge0_le_integral => //. - by apply: measurable_fun_comp => //; exact: emeasurable_funD. - by apply: emeasurable_funD; apply: measurable_fun_comp. - by move=> *; exact: lee_abs_add. by rewrite ge0_integralD //; [exact: lte_add_pinfty| exact: measurable_fun_comp|exact: measurable_fun_comp]. Qed. Lemma integrableB f g : mu_int f -> mu_int g -> mu_int (f \- g). Proof. by move=> fi gi; exact/(integrableD fi)/integrableN. Qed. Lemma integrable_add_def f : mu_int f -> \int[mu]_(x in D) f^\+ x +? - \int[mu]_(x in D) f^\- x. Proof. move=> [mf]; rewrite -[fun x => _]/(abse \o f) fune_abse => foo. rewrite ge0_integralD // in foo; last 2 first. - exact: emeasurable_fun_funepos. - exact: emeasurable_fun_funeneg. apply: ltpinfty_adde_def. - by apply: le_lt_trans foo; rewrite lee_addl// integral_ge0. - by rewrite inE (@le_lt_trans _ _ 0)// lee_oppl oppe0 integral_ge0. Qed. Lemma integrable_funepos f : mu_int f -> mu_int f^\+. Proof. move=> [Df foo]; split; first exact: emeasurable_fun_funepos. apply: le_lt_trans foo; apply: ge0_le_integral => //. - by apply/measurable_fun_comp => //; exact: emeasurable_fun_funepos. - exact/measurable_fun_comp. - by move=> t Dt; rewrite -/((abse \o f) t) fune_abse gee0_abs// lee_addl. Qed. Lemma integrable_funeneg f : mu_int f -> mu_int f^\-. Proof. move=> [Df foo]; split; first exact: emeasurable_fun_funeneg. apply: le_lt_trans foo; apply: ge0_le_integral => //. - by apply/measurable_fun_comp => //; exact: emeasurable_fun_funeneg. - exact/measurable_fun_comp. - by move=> t Dt; rewrite -/((abse \o f) t) fune_abse gee0_abs// lee_addr. Qed. Lemma integral_funeneg_lt_pinfty f : mu_int f -> \int[mu]_(x in D) f^\- x < +oo. Proof. move=> [mf]; apply: le_lt_trans; apply: ge0_le_integral => //. - by apply: emeasurable_fun_funeneg => //; exact: emeasurable_funN. - exact: measurable_fun_comp. - move=> x Dx; have [fx0|/ltW fx0] := leP (f x) 0. rewrite lee0_abs// /funeneg. by move: fx0; rewrite -{1}oppe0 -lee_oppr => /max_idPl ->. rewrite gee0_abs// /funeneg. by move: (fx0); rewrite -{1}oppe0 -lee_oppl => /max_idPr ->. Qed. Lemma integral_funepos_lt_pinfty f : mu_int f -> \int[mu]_(x in D) f^\+ x < +oo. Proof. move=> [mf]; apply: le_lt_trans; apply: ge0_le_integral => //. - by apply: emeasurable_fun_funepos => //; exact: emeasurable_funN. - exact: measurable_fun_comp. - move=> x Dx; have [fx0|/ltW fx0] := leP (f x) 0. rewrite lee0_abs// /funepos. by move: (fx0) => /max_idPr ->; rewrite -lee_oppr oppe0. by rewrite gee0_abs// /funepos; move: (fx0) => /max_idPl ->. Qed. Lemma integrable_neg_fin_num f : mu_int f -> \int[mu]_(x in D) f^\- x \is a fin_num. Proof. move=> fi. rewrite fin_numElt; apply/andP; split. by rewrite (@lt_le_trans _ _ 0) ?lte_ninfty//; exact: integral_ge0. case: fi => mf; apply: le_lt_trans; apply: ge0_le_integral => //. - exact/emeasurable_fun_funeneg. - exact/measurable_fun_comp. - by move=> x Dx; rewrite -/((abse \o f) x) (fune_abse f) lee_addr. Qed. Lemma integrable_pos_fin_num f : mu_int f -> \int[mu]_(x in D) f^\+ x \is a fin_num. Proof. move=> fi. rewrite fin_numElt; apply/andP; split. by rewrite (@lt_le_trans _ _ 0) ?lte_ninfty//; exact: integral_ge0. case: fi => mf; apply: le_lt_trans; apply: ge0_le_integral => //. - exact/emeasurable_fun_funepos. - exact/measurable_fun_comp. - by move=> x Dx; rewrite -/((abse \o f) x) (fune_abse f) lee_addl. Qed. End integrable. Notation "mu .-integrable" := (integrable mu) : type_scope. Arguments eq_integrable {d T R mu D} mD f. Section sequence_measure. Local Open Scope ereal_scope. Variables (d : _) (T : measurableType d) (R : realType). Variable m_ : {measure set T -> \bar R}^nat. Let m := mseries m_ O. Lemma integral_measure_series (D : set T) (mD : measurable D) (f : T -> \bar R) : (forall n, integrable (m_ n) D f) -> measurable_fun D f -> \sum_(n \sum_(n \int[m]_(x in D) f x = \sum_(n fi mf fmoo fpoo; rewrite integralE. rewrite ge0_integral_measure_series//; last exact/emeasurable_fun_funepos. rewrite ge0_integral_measure_series//; last exact/emeasurable_fun_funeneg. transitivity (\sum_(n n _; rewrite fineK//; [exact: integrable_pos_fin_num|exact: integrable_neg_fin_num]. have fineKn : \sum_(n n _; congr abse; rewrite fineK//. exact: integrable_neg_fin_num. have fineKp : \sum_(n n _; congr abse; rewrite fineK//. exact: integrable_pos_fin_num. rewrite nneseries_esum; last by move=> n _; exact/le0R/integral_ge0. rewrite nneseries_esum; last by move=> n _; exact/le0R/integral_ge0. rewrite -esumB//; last 4 first. - by rewrite /= /summable -nneseries_esum// -fineKp. - rewrite /summable /= -nneseries_esum. by rewrite -fineKn; exact: fmoo. by []. - by move=> n _; exact/le0R/integral_ge0. - by move=> n _; exact/le0R/integral_ge0. rewrite -summable_nneseries_esum; last first. rewrite /summable. apply: (@le_lt_trans _ _ (\esum_(i in (fun=> true)) `|(fine (\int[m_ i]_(x in D) f x))%:E|)). apply: le_esum => k _; rewrite -EFinB -fineB// -?integralE//; [exact: integrable_pos_fin_num|exact: integrable_neg_fin_num]. rewrite -nneseries_esum; last by []. apply: (@le_lt_trans _ _ (\sum_(n // n _. rewrite integralE fineB// ?EFinB. - exact: (le_trans (lee_abs_sub _ _)). - exact: integrable_pos_fin_num. - exact: integrable_neg_fin_num. apply: lte_add_pinfty; first by rewrite -fineKp. by rewrite -fineKn; exact: fmoo. by apply eq_nneseries => k _; rewrite !fineK// -?integralE//; [exact: integrable_neg_fin_num|exact: integrable_pos_fin_num]. Qed. End sequence_measure. Section integrable_lemmas. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Lemma ge0_integral_bigcup (F : (set _)^nat) (f : T -> \bar R) : (forall k, measurable (F k)) -> let D := \bigcup_k F k in mu.-integrable D f -> (forall x, D x -> 0 <= f x) -> trivIset setT F -> \int[mu]_(x in D) f x = \sum_(i mF D fi f0 tF; pose f_ N := f \_ (\big[setU/set0]_(0 <= i < N) F i). have lim_f_ t : f_ ^~ t --> (f \_ D) t. rewrite [X in _ --> X](_ : _ = ereal_sup (range (f_ ^~ t))); last first. apply/eqP; rewrite eq_le; apply/andP; split. rewrite /restrict; case: ifPn => [|_]. rewrite in_setE => -[n _ Fnt]; apply: ereal_sup_ub; exists n.+1 => //. by rewrite /f_ big_mkord patchT// in_setE big_ord_recr/=; right. rewrite (@le_trans _ _ (f_ O t))// ?ereal_sup_ub//. by rewrite /f_ patchN// big_mkord big_ord0 inE/= in_set0. apply: ub_ereal_sup => x [n _ <-]. by rewrite /f_ restrict_lee// big_mkord; exact: bigsetU_bigcup. apply: ereal_nondecreasing_cvg => a b ab. rewrite /f_ !big_mkord restrict_lee //; last exact: subset_bigsetU. by move=> x Dx; apply: f0 => //; exact: bigsetU_bigcup Dx. transitivity (\int[mu]_x lim (f_ ^~ x)). rewrite integral_mkcond; apply: eq_integral => x _. by apply/esym/cvg_lim => //; exact: lim_f_. rewrite monotone_convergence//; last 3 first. - move=> n; apply/(measurable_restrict f) => //. by apply: bigsetU_measurable => k _; exact: mF. case: fi => + _; apply/measurable_funS =>//; first exact: bigcup_measurable. by rewrite big_mkord; exact: bigsetU_bigcup. - move=> n x _; apply: erestrict_ge0 => y; rewrite big_mkord => Dy; apply: f0. exact: bigsetU_bigcup Dy. - move=> x _ a b ab; apply: restrict_lee. by move=> y; rewrite big_mkord => Dy; apply: f0; exact: bigsetU_bigcup Dy. by rewrite 2!big_mkord; apply: subset_bigsetU. transitivity (lim (fun N => \int[mu]_(x in \big[setU/set0]_(i < N) F i) f x)). congr (lim _); rewrite funeqE => n. by rewrite /f_ [in RHS]integral_mkcond big_mkord. congr (lim _); rewrite funeqE => /= n; rewrite ge0_integral_bigsetU ?big_mkord//. - case: fi => + _; apply: measurable_funS => //; first exact: bigcup_measurable. exact: bigsetU_bigcup. - by move=> y Dy; apply: f0; exact: bigsetU_bigcup Dy. - exact: sub_trivIset tF. Qed. Lemma integrableS (E D : set T) (f : T -> \bar R) : measurable E -> measurable D -> D `<=` E -> mu.-integrable E f -> mu.-integrable D f. Proof. move=> mE mD DE [mf ifoo]; split; first exact: measurable_funS mf. apply: le_lt_trans ifoo; apply: subset_integral => //. exact: measurable_fun_comp. Qed. Lemma integrable_mkcond D f : measurable D -> mu.-integrable D f <-> mu.-integrable setT (f \_ D). Proof. move=> mD; rewrite /integrable [in X in X <-> _]integral_mkcond. under [in X in X <-> _]eq_integral do rewrite restrict_abse. split => [|] [mf foo]. - by split; [exact/(measurable_restrict _ _ _ _).1| exact: le_lt_trans foo]. - by split; [exact/(measurable_restrict _ _ measurableT _).2| exact: le_lt_trans foo]. Qed. End integrable_lemmas. Arguments integrable_mkcond {d T R mu D} f. Section integrable_ae. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D) (f : T -> \bar R). Hypotheses fint : mu.-integrable D f. Lemma integrable_ae : {ae mu, forall x, D x -> f x \is a fin_num}. Proof. have [muD0|muD0] := eqVneq (mu D) 0. by exists D; split => // t /= /not_implyP[]. pose E := [set x | `|f x| = +oo /\ D x ]. have mE : measurable E. rewrite [X in measurable X](_ : _ = D `&` f @^-1` [set -oo; +oo]). by apply: fint.1 => //; exact: measurableU. rewrite predeqE => t; split=> [[/eqP ftoo Dt]|[Dt]]. split => //. by move: ftoo; rewrite /preimage /= eqe_absl => /andP[/orP[|]/eqP]; tauto. by rewrite /preimage /= => -[|]; rewrite /E /= => ->. have [ET|ET] := eqVneq E setT. have foo t : `|f t| = +oo by have [] : E t by rewrite ET. move: fint.2. suff: \int[mu]_(x in D) `|f x| = +oo by move=> ->; rewrite ltxx. by rewrite -(integral_cst_pinfty mD muD0)//; exact: eq_integral. suff: mu E = 0. move=> muE0; exists E; split => // t /= /not_implyP[Dt ftfin]; split => //. apply/eqP; rewrite eqe_absl leey andbT. by move/negP : ftfin; rewrite fin_numE negb_and 2!negbK orbC. have [->|/set0P E0] := eqVneq E set0; first by rewrite measure0. have [M M0 muM] : exists2 M, (0 <= M)%R & forall n, n%:R%:E * mu (E `&` D) <= M%:E. exists (fine (\int[mu]_(x in D) `|f x|)); first exact/le0R/integral_ge0. move=> n. rewrite -integral_indic// -ge0_integralM//; last 2 first. - apply: measurable_fun_comp=> //; apply: (@measurable_funS _ _ _ _ setT)=>//. by rewrite (_ : \1_ _ = indic_nnsfun R mE). - by move=> *; rewrite lee_fin. rewrite fineK//; last first. by case: fint => _ foo; rewrite ge0_fin_numE//; exact: integral_ge0. apply: ge0_le_integral => //. - by move=> *; rewrite lee_fin /indic. - apply/EFin_measurable_fun; apply: measurable_funM=>//. + exact: measurable_fun_cst. + apply: (@measurable_funS _ _ _ _ setT)=>//. by rewrite (_ : \1_ _ = indic_nnsfun R mE)//. - by apply: measurable_fun_comp => //; case: fint. - move=> x Dx; rewrite /= indicE. have [|xE] := boolP (x \in E); last by rewrite mule0. by rewrite /E inE /= => -[->]; rewrite leey. apply/eqP/negPn/negP => /eqP muED0. move/not_forallP : muM; apply. have [muEDoo|] := ltP (mu (E `&` D)) +oo; last first. by rewrite leye_eq => /eqP ->; exists 1%N; rewrite mul1e leye_eq. exists `|ceil (M * (fine (mu (E `&` D)))^-1)|%N.+1. apply/negP; rewrite -ltNge. rewrite -[X in _ * X](@fineK _ (mu (E `&` D))); last first. by rewrite fin_numElt muEDoo andbT (lt_le_trans _ (measure_ge0 _ _)). rewrite lte_fin -ltr_pdivr_mulr. rewrite -addn1 natrD natr_absz ger0_norm. by rewrite (le_lt_trans (ceil_ge _))// ltr_addl. by rewrite ceil_ge0// divr_ge0//; apply/le0R/measure_ge0; exact: measurableI. rewrite -lte_fin fineK. rewrite lt_neqAle measure_ge0// ?andbT. suff: E `&` D = E by move=> ->; apply/eqP/nesym. by rewrite predeqE => t; split=> -[]. by rewrite ge0_fin_numE// measure_ge0//; exact: measurableI. Qed. End integrable_ae. Section linearityM. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D) (f : T -> \bar R). Hypothesis intf : mu.-integrable D f. Lemma integralM r : \int[mu]_(x in D) (r%:E * f x) = r%:E * \int[mu]_(x in D) f x. Proof. have [r0|r0|->] := ltgtP r 0%R; last first. by under eq_fun do rewrite mul0e; rewrite mul0e integral0. - rewrite [in LHS]integralE// gt0_funeposM// gt0_funenegM//. rewrite (ge0_integralM_EFin _ _ _ _ (ltW r0)) //; last first. by apply: emeasurable_fun_funepos => //; case: intf. rewrite (ge0_integralM_EFin _ _ _ _ (ltW r0)) //; last first. by apply: emeasurable_fun_funeneg => //; case: intf. rewrite -muleBr 1?[in RHS]integralE//. by apply: integrable_add_def; case: intf. - rewrite [in LHS]integralE// lt0_funeposM// lt0_funenegM//. rewrite ge0_integralM_EFin //; last 2 first. + by apply: emeasurable_fun_funeneg => //; case: intf. + by rewrite -ler_oppr oppr0 ltW. rewrite ge0_integralM_EFin //; last 2 first. + by apply: emeasurable_fun_funepos => //; case: intf. + by rewrite -ler_oppr oppr0 ltW. rewrite -mulNe -EFinN opprK addeC EFinN mulNe -muleBr //; last first. by apply: integrable_add_def; case: intf. by rewrite [in RHS]integralE. Qed. End linearityM. Section linearity. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D) (f1 f2 : T -> R). Let g1 := EFin \o f1. Let g2 := EFin \o f2. Hypothesis if1 : mu.-integrable D g1. Hypothesis if2 : mu.-integrable D g2. Lemma integralD_EFin : \int[mu]_(x in D) (g1 \+ g2) x = \int[mu]_(x in D) g1 x + \int[mu]_(x in D) g2 x. Proof. suff: \int[mu]_(x in D) ((g1 \+ g2)^\+ x) + \int[mu]_(x in D) (g1^\- x) + \int[mu]_(x in D) (g2^\- x) = \int[mu]_(x in D) ((g1 \+ g2)^\- x) + \int[mu]_(x in D) (g1^\+ x) + \int[mu]_(x in D) (g2^\+ x). move=> h; rewrite [in LHS]integralE. move/eqP : h; rewrite -[in eqRHS]addeA [in eqRHS]addeC. have g12pos : \int[mu]_(x in D) (g1^\+ x) + \int[mu]_(x in D) (g2^\+ x) \is a fin_num. rewrite ge0_fin_numE//. by rewrite lte_add_pinfty//; exact: integral_funepos_lt_pinfty. by apply: adde_ge0; exact: integral_ge0. have g12neg : \int[mu]_(x in D) (g1^\- x) + \int[mu]_(x in D) (g2^\- x) \is a fin_num. rewrite ge0_fin_numE//. by rewrite lte_add_pinfty// ; exact: integral_funeneg_lt_pinfty. by apply: adde_ge0; exact: integral_ge0. rewrite -sube_eq; last 2 first. - rewrite ge0_fin_numE. apply: lte_add_pinfty; last exact: integral_funeneg_lt_pinfty. apply: lte_add_pinfty; last exact: integral_funeneg_lt_pinfty. have : mu.-integrable D (g1 \+ g2) by apply: integrableD. exact: integral_funepos_lt_pinfty. apply: adde_ge0; last exact: integral_ge0. by apply: adde_ge0; exact: integral_ge0. - by rewrite adde_defC fin_num_adde_def. rewrite -(addeA (\int[mu]_(x in D) (g1 \+ g2)^\+ x)). rewrite (addeC (\int[mu]_(x in D) (g1 \+ g2)^\+ x)). rewrite -addeA (addeC (\int[mu]_(x in D) g1^\- x + \int[mu]_(x in D) g2^\- x)). rewrite eq_sym -(sube_eq g12pos); last by rewrite fin_num_adde_def. move/eqP => <-. rewrite oppeD; last first. rewrite ge0_fin_numE; first exact: integral_funeneg_lt_pinfty if2. exact: integral_ge0. rewrite -addeA (addeCA (\int[mu]_(x in D) (g2^\+ x) )). by rewrite addeA -(integralE _ _ g1) -(integralE _ _ g2). have : (g1 \+ g2)^\+ \+ g1^\- \+ g2^\- = (g1 \+ g2)^\- \+ g1^\+ \+ g2^\+. rewrite funeqE => x. apply/eqP; rewrite -2!addeA [in eqRHS]addeC -sube_eq; last 2 first. by rewrite /funepos /funeneg /g1 /g2 /= !maxEFin. by rewrite /funepos /funeneg /g1 /g2 /= !maxEFin. rewrite addeAC eq_sym -sube_eq; last 2 first. by rewrite /funepos /funeneg !maxEFin. by rewrite /funepos /funeneg !maxEFin. apply/eqP. rewrite -[LHS]/((g1^\+ \+ g2^\+ \- (g1^\- \+ g2^\-)) x) -funeD_posD. by rewrite -[RHS]/((_ \- _) x) -funeD_Dpos. move/(congr1 (fun y => \int[mu]_(x in D) (y x) )). rewrite (ge0_integralD mu mD); last 4 first. - by move=> x _; rewrite adde_ge0. - apply: emeasurable_funD. by apply/emeasurable_fun_funepos/emeasurable_funD; [case: if1|case: if2]. by apply: emeasurable_fun_funeneg; case: if1. - by []. - by apply: emeasurable_fun_funeneg; case: if2. rewrite (ge0_integralD mu mD); last 4 first. - by []. - by apply/emeasurable_fun_funepos/emeasurable_funD; [case: if1|case: if2]. - by []. - by apply/emeasurable_fun_funepos/emeasurable_funN => //; case: if1. move=> ->. rewrite (ge0_integralD mu mD); last 4 first. - by move=> x _; exact: adde_ge0. - apply: emeasurable_funD. by apply/emeasurable_fun_funeneg/emeasurable_funD; [case: if1|case: if2]. by apply: emeasurable_fun_funepos; case: if1. - by []. - by apply: emeasurable_fun_funepos; case: if2. rewrite (ge0_integralD mu mD) //. - by apply/emeasurable_fun_funeneg/emeasurable_funD => //; [case: if1|case: if2]. - by apply: emeasurable_fun_funepos; case: if1. Qed. End linearity. Lemma integralB_EFin d (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}) (D : set T) (f1 f2 : T -> R) (mD : measurable D) : mu.-integrable D (EFin \o f1) -> mu.-integrable D (EFin \o f2) -> (\int[mu]_(x in D) ((f1 x)%:E - (f2 x)%:E) = (\int[mu]_(x in D) (f1 x)%:E - \int[mu]_(x in D) (f2 x)%:E))%E. Proof. move=> if1 if2; rewrite (integralD_EFin mD if1); last first. by rewrite (_ : _ \o _ = (fun x => - (f2 x)%:E))%E; [exact: integrableN|by []]. by rewrite -integralN//; exact: integrable_add_def. Qed. Lemma le_abse_integral d (R : realType) (T : measurableType d) (mu : {measure set T -> \bar R}) (D : set T) (f : T -> \bar R) (mD : measurable D) : measurable_fun D f -> (`| \int[mu]_(x in D) (f x) | <= \int[mu]_(x in D) `|f x|)%E. Proof. move=> mf. rewrite integralE (le_trans (lee_abs_sub _ _))// gee0_abs; last first. exact: integral_ge0. rewrite gee0_abs; last exact: integral_ge0. by rewrite -ge0_integralD // -?fune_abse//; [exact: emeasurable_fun_funepos | exact: emeasurable_fun_funeneg]. Qed. Section integral_indic. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Lemma integral_setI_indic (E D : set T) (mD : measurable D) (f : T -> \bar R) : measurable E -> \int[mu]_(x in E `&` D) f x = \int[mu]_(x in E) (f x * (\1_D x)%:E). Proof. move=> mE; rewrite integral_mkcondr; apply: eq_integral => x xE. by rewrite indic_restrict /patch; case: ifPn; rewrite ?mule1 ?mule0. Qed. Lemma integralEindic (D : set T) (mD : measurable D) (f : T -> \bar R) : \int[mu]_(x in D) f x = \int[mu]_(x in D) (f x * (\1_D x)%:E). Proof. by rewrite -integral_setI_indic// setIid. Qed. End integral_indic. Section ae_eq. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). Variables (mu : {measure set T -> \bar R}) (D : set T). Implicit Types f g h i : T -> \bar R. Definition ae_eq f g := {ae mu, forall x, D x -> f x = g x}. Lemma ae_eq_comp (j : \bar R -> \bar R) f g : ae_eq f g -> ae_eq (j \o f) (j \o g). Proof. move=> [N [mN N0 subN]]; exists N; split => //. by apply: subset_trans subN; apply: subsetC => x /= /[apply] ->. Qed. Lemma ae_eq_funeposneg f g : ae_eq f g <-> ae_eq f^\+ g^\+ /\ ae_eq f^\- g^\-. Proof. split=> [[N [mN N0 DfgN]]|[[A [mA A0 DfgA] [B [mB B0 DfgB]]]]]. by split; exists N; split => // x Dfgx; apply: DfgN => /=; apply: contra_not Dfgx => /= /[apply]; rewrite /funepos /funeneg => ->. exists (A `|` B); rewrite null_set_setU//; split=> //; first exact: measurableU. move=> x /= /not_implyP[Dx fgx]; apply: contrapT => /not_orP[Ax Bx]. have [fgpx|fgnx] : f^\+ x <> g^\+ x \/ f^\- x <> g^\- x. apply: contrapT => /not_orP[/contrapT fgpx /contrapT fgnx]. by apply: fgx; rewrite (funeposneg f) (funeposneg g) fgpx fgnx. - by apply: Ax; exact/DfgA/not_implyP. - by apply: Bx; exact/DfgB/not_implyP. Qed. Lemma ae_eq_sym f g : ae_eq f g -> ae_eq g f. Proof. move=> [N1 [mN1 N10 subN1]]; exists N1; split => // x /= Dba; apply: subN1 => /=. by apply: contra_not Dba => [+ Dx] => ->. Qed. Lemma ae_eq_trans f g h : ae_eq f g -> ae_eq g h -> ae_eq f h. Proof. move=> [N1 [mN1 N10 abN1]] [N2 [mN2 N20 bcN2]]; exists (N1 `|` N2); split => //. - exact: measurableU. - by rewrite null_set_setU. - rewrite -(setCK N1) -(setCK N2) -setCI; apply: subsetC => x [N1x N2x] /= Dx. move/subsetC : abN1 => /(_ _ N1x); rewrite setCK /= => ->//. by move/subsetC : bcN2 => /(_ _ N2x); rewrite setCK /= => ->. Qed. Lemma ae_eq_sub f g h i : ae_eq f g -> ae_eq h i -> ae_eq (f \- h) (g \- i). Proof. move=> [N1 [mN1 N10 abN1]] [N2 [mN2 N20 bcN2]]; exists (N1 `|` N2); split => //. - exact: measurableU. - by rewrite null_set_setU. - rewrite -(setCK N1) -(setCK N2) -setCI; apply: subsetC => x [N1x N2x] /= Dx. move/subsetC : abN1 => /(_ _ N1x); rewrite setCK /= => ->//. by move/subsetC : bcN2 => /(_ _ N2x); rewrite setCK /= => ->. Qed. Lemma ae_eq_mul2r f g h : ae_eq f g -> ae_eq (f \* h) (g \* h). Proof. move=> [N1 [mN1 N10 abN1]]; exists N1; split => // x /= /not_implyP[Dx]. move=> acbc; apply: abN1 => /=; apply/not_implyP; split => //. by apply: contra_not acbc => ->. Qed. Lemma ae_eq_mul2l f g h : ae_eq f g -> ae_eq (h \* f) (h \* g). Proof. move=> /ae_eq_mul2r-/(_ h); under eq_fun do rewrite muleC. by under [in X in ae_eq _ X -> _]eq_fun do rewrite muleC. Qed. Lemma ae_eq_mul1l f g : ae_eq f (cst 1) -> ae_eq g (g \* f). Proof. move=> /ae_eq_mul2l-/(_ g)/ae_eq_sym. by under [in X in ae_eq X _ -> _]eq_fun do rewrite mule1. Qed. Lemma ae_eq_mul f g h : ae_eq f g -> ae_eq (f \* h) (g \* h). Proof. move=> [N1 [mN1 N10 abN1]]; exists N1; split => // x /= /not_implyP[Dx]. move=> acbc; apply: abN1 => /=; apply/not_implyP; split => //. by apply: contra_not acbc => ->. Qed. Lemma ae_eq_abse f g : ae_eq f g -> ae_eq (abse \o f) (abse \o g). Proof. move=> [N [mN N0 subN]]; exists N; split => //; apply: subset_trans subN. by apply: subsetC => x /= /[apply] ->. Qed. End ae_eq. Section ae_eq_integral. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Local Notation ae_eq := (ae_eq mu). Let ae_eq_integral_abs_bounded (D : set T) (mD : measurable D) (f : T -> \bar R) M : measurable_fun D f -> (forall x, D x -> `|f x| <= M%:E) -> ae_eq D f (cst 0) -> \int[mu]_(x in D) `|f x|%E = 0. Proof. move=> mf fM [N [mA mN0 Df0N]]. pose Df_neq0 := D `&` [set x | f x != 0]. have mDf_neq0 : measurable Df_neq0 by exact: emeasurable_neq. pose f' : T -> R := indic Df_neq0. have le_f_M t : D t -> `|f t| <= M%:E * (f' t)%:E. move=> Dt; rewrite /f' indicE; have [|] := boolP (t \in Df_neq0). by rewrite inE => -[_ _]; rewrite mule1 fM. by rewrite notin_set=> /not_andP[//|/negP/negPn/eqP ->]; rewrite abse0 mule0. have : 0 <= \int[mu]_(x in D) `|f x| <= `|M|%:E * mu Df_neq0. rewrite integral_ge0//= /Df_neq0 -{2}(setIid D) setIAC -integral_indic//. rewrite -/Df_neq0 -ge0_integralM//; last 2 first. - apply: measurable_fun_comp=> //; apply: (@measurable_funS _ _ _ _ setT) => //. by rewrite (_ : \1_ _ = mindic R mDf_neq0). - by move=> x Dx; rewrite lee_fin. apply: ge0_le_integral => //. - exact: measurable_fun_comp. - by move=> x Dx; rewrite mule_ge0// lee_fin. - apply: emeasurable_funM; first exact: measurable_fun_cst. apply: measurable_fun_comp => //. apply: (@measurable_funS _ _ _ _ setT)=> //. by rewrite (_ : \1_ _ = mindic R mDf_neq0)//. - move=> x Dx. rewrite (le_trans (le_f_M _ Dx))// lee_fin /f' indicE. by case: (_ \in _) => //; rewrite ?mulr1 ?mulr0// ler_norm. have -> : mu Df_neq0 = 0. apply: (subset_measure0 _ _ _ mN0) => //. apply: subset_trans Df0N => /= x [/= f0 Dx] /=. by apply/not_implyP; split => //; exact/eqP. by rewrite mule0 -eq_le => /eqP. Qed. Lemma ae_eq_integral_abs (D : set T) (mD : measurable D) (f : T -> \bar R) : measurable_fun D f -> \int[mu]_(x in D) `|f x| = 0 <-> ae_eq D f (cst 0). Proof. move=> mf; split=> [iDf0|Df0]. exists (D `&` [set x | f x != 0]); split; [exact: emeasurable_neq| |]; last first. by move=> t /= /not_implyP [Dt /eqP ft0]. have muDf a : (0 < a)%R -> mu (D `&` [set x | a%:E <= `|f x |]) = 0. move=> a0; apply/eqP; rewrite eq_le measure_ge0 ?andbT; last first. move: (@le_integral_abse _ _ _ mu _ mD _ _ mf a0). by rewrite -lee_pdivl_mull// iDf0 mule0 setIC. rewrite [X in mu X](_ : _ = \bigcup_n (D `&` [set x | `|f x| >= n.+1%:R^-1%:E])); last first. rewrite predeqE => t; split=> [[Dt ft0]|[n _ /= [Dt nft]]]. have [ftoo|ftoo] := eqVneq `|f t| +oo%E. by exists 0%N => //; split => //=; rewrite ftoo /= leey. pose m := `|ceil (fine `|f t|)^-1|%N. have ftfin : `|f t|%E \is a fin_num. by rewrite fin_numE gt_eqF //= (lt_le_trans _ (abse_ge0 _)). exists m => //; split => //=. rewrite -(@fineK _ `|f t|) // lee_fin -ler_pinv; last 2 first. - rewrite inE unitfE fine_eq0 // abse_eq0 ft0/=; apply/lt0R. by rewrite lt_neqAle abse_ge0 -ge0_fin_numE// eq_sym abse_eq0 ft0. - by rewrite inE unitfE invr_eq0 pnatr_eq0 /= invr_gt0. rewrite invrK /m -addn1 natrD natr_absz ger0_norm ?ceil_ge0//. apply: (@le_trans _ _ ((fine `|f t|)^-1 + 1)%R); first by rewrite ler_addl. by rewrite ler_add2r// ceil_ge. by split => //; apply: contraTN nft => /eqP ->; rewrite abse0 -ltNge. transitivity (lim (fun n => mu (D `&` [set x | `|f x| >= n.+1%:R^-1%:E]))). apply/esym/cvg_lim => //; apply: cvg_mu_inc. - move=> i; apply: emeasurable_fun_c_infty => //. exact: measurable_fun_comp. - apply: bigcupT_measurable => i. by apply: emeasurable_fun_c_infty => //; exact: measurable_fun_comp. - move=> m n mn; apply/subsetPset; apply: setIS => t /=. by apply: le_trans; rewrite lee_fin lef_pinv // ?ler_nat // posrE. by rewrite (_ : (fun _ => _) = cst 0) ?lim_cst//= funeqE => n /=; rewrite muDf. pose f_ := fun n x => mine `|f x| n%:R%:E. have -> : (fun x => `|f x|) = (fun x => lim (f_^~ x)). rewrite funeqE => x; apply/esym/cvg_lim => //; apply/cvg_ballP => _/posnumP[e]. rewrite near_map; near=> n; rewrite /ball /= /ereal_ball /= /f_. have [->|fxoo] := eqVneq `|f x|%E +oo. rewrite /= (@ger0_norm _ n%:R)// ger0_norm; last first. by rewrite subr_ge0 ler_pdivr_mulr ?mul1r ?ler_addr. rewrite -{1}(@divrr _ (1 + n%:R)%R) ?unitfE; last first. by rewrite gt_eqF// {1}(_ : 1 = 1%:R)%R // -natrD add1n. rewrite -mulrBl addrK ltr_pdivr_mulr; last first. by rewrite {1}(_ : 1 = 1%:R)%R // -natrD add1n. rewrite mulrDr mulr1 ltr_spsaddl// -ltr_pdivr_mull// mulr1. near: n. exists `|ceil (1 + e%:num^-1)|%N => // n /=. rewrite -(@ler_nat R); apply: lt_le_trans. rewrite natr_absz ger0_norm ?ceil_ge ?ceil_ge0//. by rewrite (lt_le_trans _ (ceil_ge _))// ltr_addr. have fxn : `|f x| <= n%:R%:E. rewrite -(@fineK _ `|f x|); last first. by rewrite fin_numE fxoo andbT gt_eqF// (lt_le_trans _ (abse_ge0 _)). rewrite lee_fin. near: n. exists `|ceil (fine (`|f x|))|%N => // n /=. rewrite -(@ler_nat R); apply: le_trans. by rewrite natr_absz ger0_norm ?ceil_ge// ceil_ge0. by rewrite min_l// subrr normr0. transitivity (lim (fun n => \int[mu]_(x in D) (f_ n x) )). apply/esym/cvg_lim => //; apply: cvg_monotone_convergence => //. - move=> n; apply: emeasurable_fun_min => //; first exact: measurable_fun_comp. exact: measurable_fun_cst. - by move=> n t Dt; rewrite /f_ lexI abse_ge0 //= lee_fin. - move=> t Dt m n mn; rewrite /f_ lexI. have [ftm|ftm] := leP `|f t|%E m%:R%:E. by rewrite lexx /= (le_trans ftm)// lee_fin ler_nat. by rewrite (ltW ftm) /= lee_fin ler_nat. have ae_eq_f_ n : ae_eq D (f_ n) (cst 0). case: Df0 => N [mN muN0 DfN]. exists N; split => // t /= /not_implyP[Dt fnt0]. apply: DfN => /=; apply/not_implyP; split => //. apply: contra_not fnt0 => ft0. by rewrite /f_ ft0 /= normr0 min_l// lee_fin. have f_bounded n x : D x -> `|f_ n x| <= n%:R%:E. move=> Dx; rewrite /f_; have [|_] := leP `|f x| n%:R%:E. by rewrite abse_id. by rewrite gee0_abs// lee_fin. have if_0 n : \int[mu]_(x in D) `|f_ n x| = 0. apply: (@ae_eq_integral_abs_bounded _ _ _ n%:R) => //. by apply: emeasurable_fun_min => //; [exact: measurable_fun_comp|exact: measurable_fun_cst]. exact: f_bounded. rewrite (_ : (fun _ => _) = (cst 0)) // ?lim_cst// funeqE => n. rewrite (_ : (fun x => f_ n x) = abse \o f_ n); first exact: if_0. rewrite funeqE => x /=; rewrite gee0_abs// /f_. by have [|_] := leP `|f x| n%:R%:E; [by []|rewrite lee_fin]. Unshelve. all: by end_near. Qed. Lemma integral_abs_eq0 D (N : set T) (f : T -> \bar R) : measurable N -> measurable D -> N `<=` D -> measurable_fun D f -> mu N = 0 -> \int[mu]_(x in N) `|f x| = 0. Proof. move=> mN mD ND mf muN0; rewrite integralEindic//. rewrite (eq_integral (fun x => `|f x * (\1_N x)%:E|)); last first. by move=> t _; rewrite abseM (@gee0_abs _ (\1_N t)%:E)// lee_fin. apply/ae_eq_integral_abs => //. apply: emeasurable_funM => //; first exact: (@measurable_funS _ _ _ _ D). apply/EFin_measurable_fun/(@measurable_funS _ _ _ _ setT) => //. by rewrite (_ : \1_N = mindic R mN). exists N; split => // t /= /not_implyP[_]; rewrite indicE. by have [|] := boolP (t \in N); rewrite ?inE ?mule0. Qed. Lemma funID (N : set T) (mN : measurable N) (f : T -> \bar R) : let oneCN := [the {nnsfun T >-> R} of mindic R (measurableC mN)] in let oneN := [the {nnsfun T >-> R} of mindic R mN] in f = (fun x => f x * (oneCN x)%:E) \+ (fun x => f x * (oneN x)%:E). Proof. move=> oneCN oneN; rewrite funeqE => x. rewrite /oneCN /oneN/= /mindic !indicE. have [xN|xN] := boolP (x \in N). by rewrite mule1 in_setC xN mule0 add0e. by rewrite in_setC xN mule0 adde0 mule1. Qed. Lemma negligible_integrable (D N : set T) (f : T -> \bar R) : measurable N -> measurable D -> measurable_fun D f -> mu N = 0 -> mu.-integrable D f <-> mu.-integrable (D `\` N) f. Proof. move=> mN mD mf muN0. pose mCN := measurableC mN. pose oneCN : {nnsfun T >-> R} := [the {nnsfun T >-> R} of mindic R mCN]. pose oneN : {nnsfun T >-> R} := [the {nnsfun T >-> R} of mindic R mN]. have intone : mu.-integrable D (fun x => f x * (oneN x)%:E). split. apply: emeasurable_funM=> //; apply/EFin_measurable_fun. exact: (@measurable_funS _ _ _ _ setT). rewrite (eq_integral (fun x => `|f x| * (\1_N x)%:E)); last first. by move=> t _; rewrite abseM (@gee0_abs _ (\1_N t)%:E) // lee_fin. rewrite -integral_setI_indic// (@integral_abs_eq0 D)//. - exact: measurableI. - by apply: (subset_measure0 _ _ _ muN0) => //; exact: measurableI. have h1 : mu.-integrable D f <-> mu.-integrable D (fun x => f x * (oneCN x)%:E). split=> [intf|intCf]. split. apply: emeasurable_funM=> //; apply/EFin_measurable_fun => //. exact: (@measurable_funS _ _ _ _ setT). rewrite (eq_integral (fun x => `|f x| * (\1_(~` N) x)%:E)); last first. by move=> t _; rewrite abseM (@gee0_abs _ (\1_(~` N) t)%:E) // lee_fin. rewrite -integral_setI_indic//; case: intf => _; apply: le_lt_trans. by apply: subset_integral => //; [exact:measurableI|exact:measurable_fun_comp]. split => //; rewrite (funID mN f) -/oneCN -/oneN. have ? : measurable_fun D (fun x : T => f x * (oneCN x)%:E). apply: emeasurable_funM=> //. by apply/EFin_measurable_fun; exact: (@measurable_funS _ _ _ _ setT). have ? : measurable_fun D (fun x : T => f x * (oneN x)%:E). apply: emeasurable_funM => //. by apply/EFin_measurable_fun; apply: (@measurable_funS _ _ _ _ setT). apply: (@le_lt_trans _ _ (\int[mu]_(x in D) (`|f x * (oneCN x)%:E| + `|f x * (oneN x)%:E|))). apply: ge0_le_integral => //. - by apply: measurable_fun_comp => //; exact: emeasurable_funD. - by apply: emeasurable_funD; exact: measurable_fun_comp. - by move=> *; rewrite lee_abs_add. rewrite ge0_integralD//; [|exact: measurable_fun_comp|exact: measurable_fun_comp]. by apply: lte_add_pinfty; [case: intCf|case: intone]. have h2 : mu.-integrable (D `\` N) f <-> mu.-integrable D (fun x => f x * (oneCN x)%:E). split=> [intCf|intCf]. split. apply: emeasurable_funM=> //; apply/EFin_measurable_fun => //. exact: (@measurable_funS _ _ _ _ setT). rewrite (eq_integral (fun x => `|f x| * (\1_(~` N) x)%:E)); last first. by move=> t _; rewrite abseM (@gee0_abs _ (\1_(~` N) t)%:E)// lee_fin. rewrite -integral_setI_indic //; case: intCf => _; apply: le_lt_trans. apply: subset_integral=> //; [exact: measurableI|exact: measurableD|]. by apply: measurable_fun_comp => //; apply: measurable_funS mf => // ? []. split. move=> mDN A mA; rewrite setDE (setIC D) -setIA; apply: measurableI => //. exact: mf. rewrite integral_setI_indic//. case: intCf => _; rewrite (eq_integral (fun x => `|f x| * (\1_(~` N) x)%:E))//. by move=> t _; rewrite abseM (@gee0_abs _ (\1_(~` N) t)%:E)// lee_fin. by apply: (iff_trans h1); exact: iff_sym. Qed. Lemma negligible_integral (D N : set T) (f : T -> \bar R) : measurable N -> measurable D -> measurable_fun D f -> (forall x, D x -> 0 <= f x) -> mu N = 0 -> \int[mu]_(x in D) f x = \int[mu]_(x in D `\` N) f x. Proof. move=> mN mD mf f0 muN0. rewrite {1}(funID mN f) ge0_integralD//; last 4 first. - by move=> x Dx; apply: mule_ge0 => //; [exact: f0|rewrite lee_fin]. - apply: emeasurable_funM=> //; apply/EFin_measurable_fun=> //. exact: (@measurable_funS _ _ _ _ setT). - by move=> x Dx; apply: mule_ge0 => //; [exact: f0|rewrite lee_fin]. - apply: emeasurable_funM=> //; apply/EFin_measurable_fun=> //. exact: (@measurable_funS _ _ _ _ setT). rewrite -integral_setI_indic//; last exact: measurableC. rewrite -integral_setI_indic// [X in _ + X = _](_ : _ = 0) ?adde0//. rewrite (eq_integral (abse \o f)); last first. move=> x; rewrite in_setI => /andP[xD xN]. by rewrite /= gee0_abs// f0//; rewrite inE in xD. rewrite (@integral_abs_eq0 D)//; first exact: measurableI. by apply: (subset_measure0 _ _ _ muN0) => //; exact: measurableI. Qed. Lemma ge0_ae_eq_integral (D : set T) (f g : T -> \bar R) : measurable D -> measurable_fun D f -> measurable_fun D g -> (forall x, D x -> 0 <= f x) -> (forall x, D x -> 0 <= g x) -> ae_eq D f g -> \int[mu]_(x in D) (f x) = \int[mu]_(x in D) (g x). Proof. move=> mD mf mg f0 g0 [N [mN N0 subN]]. rewrite integralEindic// [RHS]integralEindic//. rewrite (negligible_integral mN)//; last 2 first. - apply: emeasurable_funM => //; apply/EFin_measurable_fun. by apply: (@measurable_funS _ _ _ _ setT) => //; rewrite (_ : \1_D = mindic R mD). - by move=> x Dx; apply: mule_ge0 => //; [exact: f0|rewrite lee_fin]. rewrite [RHS](negligible_integral mN)//; last 2 first. - apply: emeasurable_funM => //; apply/EFin_measurable_fun. by apply: (@measurable_funS _ _ _ _ setT) => //; rewrite (_ : \1_D = mindic R mD). - by move=> x Dx; apply: mule_ge0 => //; [exact: g0|rewrite lee_fin]. - apply: eq_integral => x;rewrite in_setD => /andP[_ xN]. apply: contrapT; rewrite indicE; have [|?] := boolP (x \in D). rewrite inE => Dx; rewrite !mule1. move: xN; rewrite notin_set; apply: contra_not => fxgx; apply: subN => /=. exact/not_implyP. by rewrite !mule0. Qed. Lemma ae_eq_integral (D : set T) (f g : T -> \bar R) : measurable D -> measurable_fun D f -> measurable_fun D g -> ae_eq D f g -> integral mu D f = integral mu D g. Proof. move=> mD mf mg /ae_eq_funeposneg[Dfgp Dfgn]. rewrite integralE// [in RHS]integralE//; congr (_ - _). by apply: ge0_ae_eq_integral => //; [exact: emeasurable_fun_funepos| exact: emeasurable_fun_funepos]. by apply: ge0_ae_eq_integral => //; [exact: emeasurable_fun_funeneg| exact: emeasurable_fun_funeneg]. Qed. End ae_eq_integral. Section ae_measurable_fun. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Hypothesis cmu : measure_is_complete mu. Variables (D : set T) (f g : T -> \bar R). Lemma ae_measurable_fun : ae_eq mu D f g -> measurable_fun D f -> measurable_fun D g. Proof. move=> [N [mN N0 subN]] mf B mB mD. apply: (measurability (ErealGenOInfty.measurableE R)) => // _ [_ [x ->] <-]. rewrite [X in measurable X](_ : _ = D `&` ~` N `&` (f @^-1` `]x, +oo[) `|` (D `&` N `&` g @^-1` `]x, +oo[)); last first. rewrite /preimage. apply/seteqP; split=> [y /= [Dy gyxoo]|y /= [[[Dy Ny] fyxoo]|]]. - have [->|fgy] := eqVneq (f y) (g y). have [yN|yN] := boolP (y \in N). by right; split => //; rewrite inE in yN. by left; split => //; rewrite notin_set in yN. by right; split => //; split => //; apply: subN => /= /(_ Dy); exact/eqP. - split => //; have [<-//|fgy] := eqVneq (f y) (g y). by exfalso; apply/Ny/subN => /= /(_ Dy); exact/eqP. - by move=> [[]]. apply: measurableU. - rewrite setIAC; apply: measurableI; last exact/measurableC. exact/mf/emeasurable_itv_bnd_pinfty. - by apply: cmu; exists N; split => //; rewrite setIAC; apply: subIset; right. Qed. End ae_measurable_fun. Section integralD. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D) (f1 f2 : T -> \bar R). Hypotheses (if1 : mu.-integrable D f1) (if2 : mu.-integrable D f2). Lemma integralD : \int[mu]_(x in D) (f1 x + f2 x) = \int[mu]_(x in D) f1 x + \int[mu]_(x in D) f2 x. Proof. pose A := D `&` [set x | f1 x \is a fin_num]. pose B := D `&` [set x | f2 x \is a fin_num]. have mA : measurable A by apply: emeasurable_fin_num => //; case: if1. have mB : measurable B by apply: emeasurable_fin_num => //; case: if2. have mAB : measurable (A `&` B) by apply: measurableI. pose g1 := (fine \o f1 \_ (A `&` B))%R. pose g2 := (fine \o f2 \_ (A `&` B))%R. have ig1 : mu.-integrable D (EFin \o g1). rewrite (_ : _ \o _ = f1 \_ (A `&` B)) //. apply: (integrableS measurableT)=>//; apply/(integrable_mkcond _ _).1 => //. by apply: integrableS if1=>//; rewrite -setIAC -setIA; apply: subIset; left. rewrite /g1 funeqE => x //=; rewrite !/restrict; case: ifPn => //. rewrite 2!in_setI => /andP[/andP[xA f1xfin] _] /=. by rewrite fineK//; rewrite inE in f1xfin. have ig2 : mu.-integrable D (EFin \o g2). rewrite (_ : _ \o _ = f2 \_ (A `&` B)) //. apply: (integrableS measurableT)=>//; apply/(integrable_mkcond _ _).1 => //. by apply: integrableS if2=>//; rewrite -setIAC -setIA; apply: subIset; left. rewrite /g2 funeqE => x //=; rewrite !/restrict; case: ifPn => //. rewrite in_setI => /andP[_]; rewrite in_setI => /andP[xB f2xfin] /=. by rewrite fineK//; rewrite inE in f2xfin. transitivity (\int[mu]_(x in D) (EFin \o (g1 \+ g2)%R) x). apply: ae_eq_integral => //. - by apply: emeasurable_funD => //; [case: if1|case: if2]. - rewrite (_ : _ \o _ = (EFin \o g1) \+ (EFin \o g2))//. by apply: emeasurable_funD => //; [case: ig1|case: ig2]. - have [N1 [mN1 N10 subN1]] := integrable_ae mD if1. have [N2 [mN2 N20 subN2]] := integrable_ae mD if2. exists (N1 `|` N2); split; [exact: measurableU|by rewrite null_set_setU|]. rewrite -(setCK N1) -(setCK N2) -setCI. apply: subsetC => x [N1x N2x] /= Dx. move/subsetC : subN1 => /(_ x N1x); rewrite setCK /= => /(_ Dx) f1x. move/subsetC : subN2 => /(_ x N2x); rewrite setCK /= => /(_ Dx) f2x. rewrite /g1 /g2 /restrict /=; have [|] := boolP (x \in A `&` B). by rewrite in_setI => /andP[xA xB] /=; rewrite EFinD !fineK. by rewrite in_setI negb_and => /orP[|]; rewrite in_setI negb_and /= (mem_set Dx)/= notin_set/=. - rewrite (_ : _ \o _ = (EFin \o g1) \+ (EFin \o g2))// integralD_EFin//. congr (_ + _). + apply: ae_eq_integral => //; [by case: ig1|by case: if1|]. have [N1 [mN1 N10 subN1]] := integrable_ae mD if1. have [N2 [mN2 N20 subN2]] := integrable_ae mD if2. exists (N1 `|` N2); split; [exact: measurableU|by rewrite null_set_setU|]. rewrite -(setCK N1) -(setCK N2) -setCI. apply: subsetC => x [N1x N2x] /= Dx. move/subsetC : subN1 => /(_ x N1x); rewrite setCK /= => /(_ Dx) f1x. move/subsetC : subN2 => /(_ x N2x); rewrite setCK /= => /(_ Dx) f2x. rewrite /g1 /= /restrict. have [/=|] := boolP (x \in A `&` B); first by rewrite fineK. by rewrite in_setI negb_and => /orP[|]; rewrite in_setI negb_and /= (mem_set Dx) /= notin_set. + apply: ae_eq_integral => //;[by case: ig2|by case: if2|]. have [N1 [mN1 N10 subN1]] := integrable_ae mD if1. have [N2 [mN2 N20 subN2]] := integrable_ae mD if2. exists (N1 `|` N2); split; [exact: measurableU|by rewrite null_set_setU|]. rewrite -(setCK N1) -(setCK N2) -setCI. apply: subsetC => x [N1x N2x] /= Dx. move/subsetC : subN1 => /(_ x N1x); rewrite setCK /= => /(_ Dx) f1x. move/subsetC : subN2 => /(_ x N2x); rewrite setCK /= => /(_ Dx) f2x. rewrite /g2 /= /restrict. have [/=|] := boolP (x \in A `&` B); first by rewrite fineK. by rewrite in_setI negb_and => /orP[|]; rewrite in_setI negb_and /= (mem_set Dx) /= notin_set. Qed. End integralD. Section integralB. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType). Variables (mu : {measure set T -> \bar R}) (D : set T). Variables (mD : measurable D) (f1 f2 : T -> \bar R). Hypotheses (if1 : mu.-integrable D f1) (if2 : mu.-integrable D f2). Lemma integralB : \int[mu]_(x in D) (f1 \- f2) x = \int[mu]_(x in D) f1 x - \int[mu]_(x in D) f2 x. Proof. rewrite -[in RHS](@integralN _ _ _ _ _ f2); last exact: integrable_add_def. by rewrite -[in RHS]integralD//; exact: integrableN. Qed. End integralB. Section integral_counting. Local Open Scope ereal_scope. Variables (R : realType). Lemma counting_dirac (A : set nat) : counting R A = \sum_(n : \sum_(n \bar R. rewrite nneseries_esum// (_ : [set _ | _] = setT); last exact/seteqP. rewrite [in LHS](esumID A)// !setTI [X in _ + X](_ : _ = 0) ?adde0//. by apply esum0 => i Ai; rewrite /= /dirac indicE memNset. rewrite /counting/=; case: ifPn => /asboolP finA. by rewrite -finite_card_dirac. by rewrite infinite_card_dirac. Qed. Lemma summable_integral_dirac (a : nat -> \bar R) : summable setT a -> \sum_(n sa. apply: (@le_lt_trans _ _ (\sum_(i // n _; rewrite integral_dirac//. move: (@summable_pinfty _ _ _ _ sa n Logic.I). by case: (a n) => //= r _; rewrite indicE/= mem_set// mul1r. move: (sa); rewrite /summable (_ : [set: nat] = (fun=> true))//; last exact/seteqP. rewrite -nneseries_esum//; apply: le_lt_trans. by apply lee_nneseries => // n _ /=; case: (a n) => //; rewrite leey. Qed. Lemma integral_count (a : nat -> \bar R) : summable setT a -> \int[counting R]_t (a t) = \sum_(k sa. transitivity (\int[mseries (fun n => [the measure _ _ of \d_ n]) O]_t a t). congr (integral _ _ _); apply/funext => A. by rewrite /= counting_dirac. rewrite (@integral_measure_series _ _ R (fun n => [the measure _ _ of \d_ n]) setT)//=. - apply: eq_nneseries => i _; rewrite integral_dirac//=. by rewrite indicE mem_set// mul1e. - move=> n; split; first by []. by rewrite integral_dirac//= indicE mem_set// mul1e; exact: (summable_pinfty sa). - by apply: summable_integral_dirac => //; exact: summable_funeneg. - by apply: summable_integral_dirac => //; exact: summable_funepos. Qed. End integral_counting. Section subadditive_countable. Local Open Scope ereal_scope. Variables (d : _) (T : measurableType d) (R : realType). Variable (mu : {measure set T -> \bar R}). Lemma integrable_abse (D : set T) : measurable D -> forall f : T -> \bar R, mu.-integrable D f -> mu.-integrable D (abse \o f). Proof. move=> mD f [mf fi]; split; first exact: measurable_fun_comp. apply: le_lt_trans fi; apply: ge0_le_integral => //. - by apply: measurable_fun_comp => //; exact: measurable_fun_comp. - exact: measurable_fun_comp. - by move=> t Dt //=; rewrite abse_id. Qed. Lemma integrable_summable (F : (set T)^nat) (g : T -> \bar R): trivIset setT F -> (forall k, measurable (F k)) -> mu.-integrable (\bigcup_k F k) g -> summable [set: nat] (fun i => \int[mu]_(x in F i) g x). Proof. move=> tF mF fi. rewrite /summable -(_ : [set _ | true] = setT); last exact/seteqP. rewrite -nneseries_esum//. case: (fi) => _; rewrite ge0_integral_bigcup//; last first. by apply: integrable_abse => //; exact: bigcup_measurable. apply: le_lt_trans; apply: lee_lim. - exact: is_cvg_ereal_nneg_natsum_cond. - by apply: is_cvg_ereal_nneg_natsum_cond => n _ _; exact: integral_ge0. - apply: nearW => n; apply: lee_sum => m _; apply: le_abse_integral => //. by apply: measurable_funS fi.1 => //; [exact: bigcup_measurable| exact: bigcup_sup]. Qed. Lemma integral_bigcup (F : (set _)^nat) (g : T -> \bar R) : trivIset setT F -> (forall k, measurable (F k)) -> mu.-integrable (\bigcup_k F k) g -> (\int[mu]_(x in \bigcup_i F i) g x = \sum_(i tF mF fi. have ? : \int[mu]_(x in \bigcup_i F i) g x \is a fin_num. rewrite fin_numElt -(lte_absl _ +oo). apply: le_lt_trans fi.2; apply: le_abse_integral => //. exact: bigcupT_measurable. exact: fi.1. transitivity (\int[mu]_(x in \bigcup_i F i) g^\+ x - \int[mu]_(x in \bigcup_i F i) g^\- x)%E. rewrite -integralB; last 3 first. - exact: bigcupT_measurable. - by apply: integrable_funepos => //; exact: bigcupT_measurable. -by apply: integrable_funeneg => //; exact: bigcupT_measurable. by apply eq_integral => t Ft; rewrite [in LHS](funeposneg g). transitivity (\sum_(i // i; rewrite [RHS]integralE. transitivity ((\sum_(i //; exact: bigcupT_measurable. by rewrite ge0_integral_bigcup//; apply: integrable_funepos => //; [exact: bigcupT_measurable|exact: integrableN]. rewrite [X in X - _]nneseries_esum; last by move=> n _; exact: integral_ge0. rewrite [X in _ - X]nneseries_esum; last by move=> n _; exact: integral_ge0. rewrite set_true -esumB//=; last 4 first. - apply: integrable_summable => //; apply: integrable_funepos => //. exact: bigcup_measurable. - apply: integrable_summable => //; apply: integrable_funepos => //. exact: bigcup_measurable. - exact: integrableN. - by move=> n _; exact: integral_ge0. - by move=> n _; exact: integral_ge0. rewrite summable_nneseries; last first. rewrite (_ : (fun i : nat => _) = (fun i => \int[mu]_(x in F i) g x)); last first. by apply/funext => i; rewrite -integralE. rewrite -(_ : [set: nat] = (fun=> true)); last exact/seteqP. exact: integrable_summable. by congr (_ - _)%E; rewrite nneseries_esum// set_true. Qed. End subadditive_countable. Section dominated_convergence_lemma. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D) (f_ : (T -> \bar R)^nat). Variables (f : T -> \bar R) (g : T -> \bar R). Hypothesis mf_ : forall n, measurable_fun D (f_ n). Hypothesis f_f : forall x, D x -> f_ ^~ x --> f x. Hypothesis fing : forall x, D x -> g x \is a fin_num. Hypothesis ig : mu.-integrable D g. Hypothesis absfg : forall n x, D x -> `|f_ n x| <= g x. Let g0 x : D x -> 0 <= g x. Proof. by move=> Dx; rewrite (le_trans _ (@absfg O _ Dx))// lee_fin. Qed. Let mf : measurable_fun D f. Proof. exact: (emeasurable_fun_cvg _ _ mf_ f_f). Qed. Local Lemma dominated_integrable : mu.-integrable D f. Proof. split => //; have Dfg x : D x -> `| f x | <= g x. move=> Dx; have /(@cvg_lim _) <- // : `|f_ n x| @[n --> \oo] --> `|f x|. by apply: cvg_abse => //; exact: f_f. apply: ereal_lim_le => //. - by apply: is_cvg_abse; apply/cvg_ex; eexists; exact: f_f. - by apply: nearW => n; exact: absfg. move: ig => [mg]; apply: le_lt_trans; apply: ge0_le_integral => //. - exact: measurable_fun_comp. - exact: measurable_fun_comp. - by move=> x Dx /=; rewrite (gee0_abs (g0 Dx)); exact: Dfg. Qed. Let g_ n x := `|f_ n x - f x|. Let cvg_g_ x : D x -> g_ ^~ x --> 0. Proof. move=> Dx; rewrite -abse0; apply: cvg_abse. move: (f_f Dx); case: (f x) => [r|/=|/=]. - by move=> f_r; apply/ereal_cvg_sub0. - have gx1 : (0 < fine (g x) + 1)%R. by rewrite (@le_lt_trans _ _ (fine (g x))) ?ltr_addl//; exact/le0R/g0. move/ereal_cvgPpinfty/(_ _ gx1) => [n _]/(_ _ (leqnn n)) h. have : (fine (g x) + 1)%:E <= g x. by rewrite (le_trans h)// (le_trans _ (absfg n Dx))// lee_abs. by case: (g x) (fing Dx) => [r _| |]//; rewrite leNgt EFinD lte_addl. - have gx1 : (- (fine (g x) + 1) < 0)%R. by rewrite ltr_oppl oppr0 ltr_spaddr//; exact/le0R/g0. move/ereal_cvgPninfty/(_ _ gx1) => [n _]/(_ _ (leqnn n)) h. have : (fine (g x) + 1)%:E <= g x. move: h; rewrite EFinN lee_oppr => /le_trans ->//. by rewrite (le_trans _ (absfg n Dx))// -abseN lee_abs. by case: (g x) (fing Dx) => [r _| |]//; rewrite leNgt EFinD lte_addl. Qed. Let gg_ n x : D x -> 0 <= 2%:E * g x - g_ n x. Proof. move=> Dx; rewrite subre_ge0; last by rewrite fin_numM// fing. rewrite -(fineK (fing Dx)) -EFinM mulr_natl mulr2n /g_. rewrite (le_trans (lee_abs_sub _ _))// [in leRHS]EFinD lee_add//. by rewrite fineK// ?fing// absfg. have f_fx : `|(f_ n x)| @[n --> \oo] --> `|f x| by apply: cvg_abse; exact: f_f. move/cvg_lim : (f_fx) => <-//. apply: ereal_lim_le; first by apply/cvg_ex; eexists; exact: f_fx. by apply: nearW => k; rewrite fineK ?fing//; apply: absfg. Qed. Let mgg n : measurable_fun D (fun x => 2%:E * g x - g_ n x). Proof. apply/emeasurable_funB => //; first by apply: emeasurable_funeM; case: ig. by apply/measurable_fun_comp => //; exact: emeasurable_funB. Qed. Let gg_ge0 n x : D x -> 0 <= 2%:E * g x - g_ n x. Proof. by move=> Dx; rewrite gg_. Qed. Local Lemma dominated_cvg0 : (fun n => \int[mu]_(x in D) g_ n x) --> 0. Proof. have := fatou mu mD mgg gg_ge0. rewrite [X in X <= _ -> _](_ : _ = \int[mu]_(x in D) (2%:E * g x) ); last first. apply: eq_integral => t; rewrite inE => Dt. rewrite elim_inf_shift//; last by rewrite fin_numM// fing. rewrite is_cvg_elim_infE//; last first. by apply: ereal_is_cvgN; apply/cvg_ex; eexists; exact: cvg_g_. rewrite [X in _ + X](_ : _ = 0) ?adde0//; apply/cvg_lim => //. by rewrite -(oppe0); apply: ereal_cvgN; exact: cvg_g_. have i2g : \int[mu]_(x in D) (2%:E * g x) < +oo. rewrite integralM// lte_mul_pinfty// ?lee_fin//; case: ig => _. apply: le_lt_trans; rewrite le_eqVlt; apply/orP; left; apply/eqP. by apply: eq_integral => t Dt; rewrite gee0_abs// g0//; rewrite inE in Dt. have ? : \int[mu]_(x in D) (2%:E * g x) \is a fin_num. by rewrite ge0_fin_numE// integral_ge0// => x Dx; rewrite mule_ge0 ?lee_fin ?g0. rewrite [X in _ <= X -> _](_ : _ = \int[mu]_(x in D) (2%:E * g x) + - elim_sup (fun n => \int[mu]_(x in D) g_ n x)); last first. rewrite (_ : (fun _ => _) = (fun n => \int[mu]_(x in D) (2%:E * g x) + \int[mu]_(x in D) - g_ n x)); last first. rewrite funeqE => n; rewrite integralB//. - by rewrite -integral_ge0N// => x Dx//; rewrite /g_. - exact: integrablerM. - have integrable_normfn : mu.-integrable D (abse \o f_ n). apply: le_integrable ig => //. - exact: measurable_fun_comp. - by move=> x Dx /=; rewrite abse_id (le_trans (absfg _ Dx))// lee_abs. suff: mu.-integrable D (fun x => `|f_ n x| + `|f x|). apply: le_integrable => //. - by apply: measurable_fun_comp => //; exact: emeasurable_funB. - move=> x Dx. by rewrite /g_ abse_id (le_trans (lee_abs_sub _ _))// lee_abs. apply: integrableD; [by []| by []|]. apply: le_integrable dominated_integrable => //. - exact: measurable_fun_comp. - by move=> x Dx; rewrite /= abse_id. rewrite elim_inf_shift // -elim_infN; congr (_ + elim_inf _). by rewrite funeqE => n /=; rewrite -integral_ge0N// => x Dx; rewrite /g_. rewrite addeC -lee_subl_addr// subee// lee_oppr oppe0 => lim_ge0. by apply/elim_sup_le_cvg => // n; rewrite integral_ge0// => x _; rewrite /g_. Qed. Local Lemma dominated_cvg : (fun n => \int[mu]_(x in D) f_ n x) --> \int[mu]_(x in D) f x. Proof. have h n : `| \int[mu]_(x in D) f_ n x - \int[mu]_(x in D) f x | <= \int[mu]_(x in D) g_ n x. rewrite -(integralB _ _ dominated_integrable)//; last first. by apply: le_integrable ig => // x Dx /=; rewrite (gee0_abs (g0 Dx)) absfg. by apply: le_abse_integral => //; exact: emeasurable_funB. suff: (fun n => `| \int[mu]_(x in D) f_ n x - \int[mu]_(x in D) f x |) --> 0. move/ereal_cvg_abs0/ereal_cvg_sub0; apply. rewrite fin_numElt (_ : -oo = - +oo)// -lte_absl. case: dominated_integrable => ?; apply: le_lt_trans. by apply: (le_trans _ (@le_abse_integral _ _ _ mu D f mD _)). apply: (@ereal_squeeze _ (cst 0) _ (fun n => \int[mu]_(x in D) g_ n x)). - by apply: nearW => n; rewrite abse_ge0//=; exact: h. - exact: cvg_cst. - exact: dominated_cvg0. Qed. End dominated_convergence_lemma. Arguments dominated_integrable {d T R mu D} _ f_ f g. Section dominated_convergence_theorem. Local Open Scope ereal_scope. Variables (d : measure_display) (T : measurableType d) (R : realType) (mu : {measure set T -> \bar R}). Variables (D : set T) (mD : measurable D). Variables (f_ : (T -> \bar R)^nat) (f : T -> \bar R) (g : T -> \bar R). Hypothesis mf_ : forall n, measurable_fun D (f_ n). Hypothesis mf : measurable_fun D f. Hypothesis f_f : {ae mu, forall x, D x -> f_ ^~ x --> f x}. Hypothesis ig : mu.-integrable D g. Hypothesis f_g : {ae mu, forall x n, D x -> `|f_ n x| <= g x}. Let g_ n x := `|f_ n x - f x|. Theorem dominated_convergence : [/\ mu.-integrable D f, (fun n => \int[mu]_(x in D) (g_ n x) ) --> 0 & (fun n => \int[mu]_(x in D) (f_ n x) ) --> \int[mu]_(x in D) (f x) ]. Proof. have [N1 [mN1 N10 subN1]] := f_f. have [N2 [mN2 N20 subN2]] := f_g. have [N3 [mN3 N30 subN3]] := integrable_ae mD ig. pose N := N1 `|` N2 `|` N3. have mN : measurable N by apply: measurableU => //; exact: measurableU. have N0 : mu N = 0. by rewrite null_set_setU// ?null_set_setU//; exact: measurableU. pose f' := f \_ (D `\` N); pose g' := g \_ (D `\` N). pose f_' := fun n => f_ n \_ (D `\` N). have f_f' x : D x -> f_' ^~ x --> f' x. move=> Dx; rewrite /f_' /f' /restrict in_setD mem_set//=. have [/= xN|/= xN] := boolP (x \in N); first exact: cvg_cst. apply: contraPP (xN) => h; apply/negP; rewrite negbK inE; left; left. by apply: subN1 => /= /(_ Dx); exact: contra_not h. have f_g' n x : D x -> `|f_' n x| <= g' x. move=> Dx; rewrite /f_' /g' /restrict in_setD mem_set//=. have [/=|/= xN] := boolP (x \in N); first by rewrite normr0. apply: contrapT => fg; move: xN; apply/negP; rewrite negbK inE; left; right. by apply: subN2 => /= /(_ n Dx). have ? : measurable_fun D (\1_(D `\` N) : T -> R). apply: (@measurable_funS _ _ _ _ setT) => //. by rewrite (_ : \1_ _ = mindic R (measurableD mD mN)). have mu_ n : measurable_fun D (f_' n). apply/(measurable_restrict (f_ n) (measurableD mD mN) _ _).1 => //. by apply: measurable_funS (mf_ _) => // x []. have ig' : mu.-integrable D g'. apply: (integrableS measurableT) => //. apply/(integrable_mkcond g (measurableD mD mN)).1. by apply: integrableS ig => //; exact: measurableD. have finv x : D x -> g' x \is a fin_num. move=> Dx; rewrite /g' /restrict in_setD// mem_set//=. have [//|xN /=] := boolP (x \in N). apply: contrapT => fing; move: xN; apply/negP; rewrite negbK inE; right. by apply: subN3 => /= /(_ Dx). split. - have if' : mu.-integrable D f' by exact: (dominated_integrable _ f_' _ g'). split => //. move: if' => [?]; apply: le_lt_trans. rewrite le_eqVlt; apply/orP; left; apply/eqP/ae_eq_integral => //; [exact: measurable_fun_comp|exact: measurable_fun_comp|]. exists N; split => //; rewrite -(setCK N); apply: subsetC => x Nx Dx. by rewrite /f' /restrict mem_set. - have := @dominated_cvg0 _ _ _ _ _ mD _ _ _ mu_ f_f' finv ig' f_g'. set X := (X in _ -> X --> _); rewrite [X in X --> _ -> _](_ : _ = X) //. apply/funext => n; apply: ae_eq_integral => //. + apply: measurable_fun_comp => //; apply: emeasurable_funB => //. apply/(measurable_restrict _ (measurableD _ _) _ _).1 => //. by apply: (@measurable_funS _ _ _ _ D) => // x []. + by rewrite /g_; apply: measurable_fun_comp => //; exact: emeasurable_funB. + exists N; split => //; rewrite -(setCK N); apply: subsetC => x /= Nx Dx. by rewrite /f_' /f' /restrict mem_set. - have := @dominated_cvg _ _ _ _ _ mD _ _ _ mu_ f_f' finv ig' f_g'. set X := (X in _ -> X --> _); rewrite [X in X --> _ -> _](_ : _ = X) //; last first. apply/funext => n; apply ae_eq_integral => //. exists N; split => //; rewrite -(setCK N); apply: subsetC => x /= Nx Dx. by rewrite /f_' /restrict mem_set. set Y := (X in _ -> _ --> X); rewrite [X in _ --> X -> _](_ : _ = Y) //. apply: ae_eq_integral => //. apply/(measurable_restrict _ (measurableD _ _) _ _).1 => //. by apply: (@measurable_funS _ _ _ _ D) => // x []. exists N; split => //; rewrite -(setCK N); apply: subsetC => x /= Nx Dx. by rewrite /f' /restrict mem_set. Qed. End dominated_convergence_theorem. (******************************************************************************) (* * product measure *) (******************************************************************************) Section measurable_section. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2). Implicit Types (A : set (T1 * T2)). Lemma mem_set_pair1 x y A : (y \in [set y' | (x, y') \in A]) = ((x, y) \in A). Proof. by apply/idP/idP => [|]; [rewrite inE|rewrite !inE /= inE]. Qed. Lemma mem_set_pair2 x y A : (x \in [set x' | (x', y) \in A]) = ((x, y) \in A). Proof. by apply/idP/idP => [|]; [rewrite inE|rewrite 2!inE /= inE]. Qed. Variable R : realType. Lemma measurable_xsection A x : measurable A -> measurable (xsection A x). Proof. move=> mA. pose f : T1 * T2 -> \bar R := EFin \o indic_nnsfun R mA. have mf : measurable_fun setT f by apply/EFin_measurable_fun/measurable_funP. have _ : (fun y => (y \in xsection A x)%:R%:E) = f \o (fun y => (x, y)). rewrite funeqE => y /=; rewrite /xsection /f. by rewrite /= /mindic indicE/= mem_set_pair1. have -> : xsection A x = (fun y => f (x, y)) @^-1` [set 1%E]. rewrite predeqE => y; split; rewrite /xsection /preimage /= /f. by rewrite /= /mindic indicE/= => ->. rewrite /= /mindic indicE. by case: (_ \in _) => //= -[] /eqP; rewrite eq_sym oner_eq0. by rewrite -(setTI (_ @^-1` _)); exact: measurable_fun_prod1. Qed. Lemma measurable_ysection A y : measurable A -> measurable (ysection A y). Proof. move=> mA. pose f : T1 * T2 -> \bar R := EFin \o indic_nnsfun R mA. have mf : measurable_fun setT f by apply/EFin_measurable_fun/measurable_funP. have _ : (fun x => (x \in ysection A y)%:R%:E) = f \o (fun x => (x, y)). rewrite funeqE => x /=; rewrite /ysection /f. by rewrite /= /mindic indicE mem_set_pair2. have -> : ysection A y = (fun x => f (x, y)) @^-1` [set 1%E]. rewrite predeqE => x; split; rewrite /ysection /preimage /= /f. by rewrite /= /mindic indicE => ->. rewrite /= /mindic indicE. by case: (_ \in _) => //= -[] /eqP; rewrite eq_sym oner_eq0. by rewrite -(setTI (_ @^-1` _)); exact: measurable_fun_prod2. Qed. End measurable_section. Section ndseq_closed_B. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Implicit Types A : set (T1 * T2). Section xsection. Variables (pt2 : T2) (m2 : {measure set T2 -> \bar R}). Let phi A := m2 \o xsection A. Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. Lemma xsection_ndseq_closed : ndseq_closed B. Proof. move=> F ndF; rewrite /B /= => BF; split. by apply: bigcupT_measurable => n; have [] := BF n. have phiF x : (fun i => phi (F i) x) --> phi (\bigcup_i F i) x. rewrite /phi /= xsection_bigcup; apply: cvg_mu_inc => //. - by move=> n; apply: measurable_xsection; case: (BF n). - by apply: bigcupT_measurable => i; apply: measurable_xsection; case: (BF i). - move=> m n mn; apply/subsetPset => y; rewrite /xsection/= !inE. by have /subsetPset FmFn := ndF _ _ mn; exact: FmFn. apply: (emeasurable_fun_cvg (phi \o F)) => //. - by move=> i; have [] := BF i. - by move=> x _; exact: phiF. Qed. End xsection. Section ysection. Variables (m1 : {measure set T1 -> \bar R}). Let psi A := m1 \o ysection A. Let B := [set A | measurable A /\ measurable_fun setT (psi A)]. Lemma ysection_ndseq_closed : ndseq_closed B. Proof. move=> F ndF; rewrite /B /= => BF; split. by apply: bigcupT_measurable => n; have [] := BF n. have psiF x : (fun i => psi (F i) x) --> psi (\bigcup_i F i) x. rewrite /psi /= ysection_bigcup; apply: cvg_mu_inc => //. - by move=> n; apply: measurable_ysection; case: (BF n). - by apply: bigcupT_measurable => i; apply: measurable_ysection; case: (BF i). - move=> m n mn; apply/subsetPset => y; rewrite /ysection/= !inE. by have /subsetPset FmFn := ndF _ _ mn; exact: FmFn. apply: (emeasurable_fun_cvg (psi \o F)) => //. - by move=> i; have [] := BF i. - by move=> x _; exact: psiF. Qed. End ysection. End ndseq_closed_B. Section measurable_prod_subset. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Implicit Types A : set (T1 * T2). Section xsection. Variable (m2 : {measure set T2 -> \bar R}) (D : set T2) (mD : measurable D). Let m2D := mrestr m2 mD. HB.instance Definition _ := Measure.on m2D. Let phi A := m2D \o xsection A. Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. Lemma measurable_prod_subset_xsection (m2D_bounded : exists M, forall X, measurable X -> (m2D X < M%:E)%E) : measurable `<=` B. Proof. rewrite measurable_prod_measurableType. set C := [set A1 `*` A2 | A1 in measurable & A2 in measurable]. have CI : setI_closed C. move=> X Y [X1 mX1 [X2 mX2 <-{X}]] [Y1 mY1 [Y2 mY2 <-{Y}]]. exists (X1 `&` Y1); first exact: measurableI. by exists (X2 `&` Y2); [exact: measurableI|rewrite setMI]. have CT : C setT by exists setT => //; exists setT => //; rewrite setMTT. have CB : C `<=` B. move=> X [X1 mX1 [X2 mX2 <-{X}]]; split; first exact: measurableM. have -> : phi (X1 `*` X2) = (fun x => m2D X2 * (\1_X1 x)%:E)%E. rewrite funeqE => x; rewrite indicE /phi /m2/= /mrestr. have [xX1|xX1] := boolP (x \in X1); first by rewrite mule1 in_xsectionM. by rewrite mule0 notin_xsectionM// set0I measure0. apply: emeasurable_funeM => //; apply/EFin_measurable_fun. by rewrite (_ : \1_ _ = mindic R mX1). suff monoB : monotone_class setT B by exact: monotone_class_subset. split => //; [exact: CB| |exact: xsection_ndseq_closed]. move=> X Y XY [mX mphiX] [mY mphiY]; split; first exact: measurableD. have -> : phi (X `\` Y) = (fun x => phi X x - phi Y x)%E. rewrite funeqE => x; rewrite /phi/= xsectionD// /m2D measureD. - by rewrite setIidr//; exact: le_xsection. - exact: measurable_xsection. - exact: measurable_xsection. - move: m2D_bounded => [M m2M]. rewrite (lt_le_trans (m2M (xsection X x) _))// ?leey//. exact: measurable_xsection. exact: emeasurable_funB. Qed. End xsection. Section ysection. Variable (m1 : {measure set T1 -> \bar R}) (D : set T1) (mD : measurable D). Let m1D := mrestr m1 mD. HB.instance Definition _ := Measure.on m1D. Let psi A := m1D \o ysection A. Let B := [set A | measurable A /\ measurable_fun setT (psi A)]. Lemma measurable_prod_subset_ysection (m1_bounded : exists M, forall X, measurable X -> (m1D X < M%:E)%E) : measurable `<=` B. Proof. rewrite measurable_prod_measurableType. set C := [set A1 `*` A2 | A1 in measurable & A2 in measurable]. have CI : setI_closed C. move=> X Y [X1 mX1 [X2 mX2 <-{X}]] [Y1 mY1 [Y2 mY2 <-{Y}]]. exists (X1 `&` Y1); first exact: measurableI. by exists (X2 `&` Y2); [exact: measurableI|rewrite setMI]. have CT : C setT by exists setT => //; exists setT => //; rewrite setMTT. have CB : C `<=` B. move=> X [X1 mX1 [X2 mX2 <-{X}]]; split; first exact: measurableM. have -> : psi (X1 `*` X2) = (fun x => m1D X1 * (\1_X2 x)%:E)%E. rewrite funeqE => y; rewrite indicE /psi /m1/= /mrestr. have [yX2|yX2] := boolP (y \in X2); first by rewrite mule1 in_ysectionM. by rewrite mule0 notin_ysectionM// set0I measure0. apply: emeasurable_funeM => //; apply/EFin_measurable_fun. by rewrite (_ : \1_ _ = mindic R mX2). suff monoB : monotone_class setT B by exact: monotone_class_subset. split => //; [exact: CB| |exact: ysection_ndseq_closed]. move=> X Y XY [mX mphiX] [mY mphiY]; split; first exact: measurableD. have -> : psi (X `\` Y) = (fun x => psi X x - psi Y x)%E. rewrite funeqE => y; rewrite /psi/= ysectionD// /m1D measureD. - by rewrite setIidr//; exact: le_ysection. - exact: measurable_ysection. - exact: measurable_ysection. - move: m1_bounded => [M m1M]. rewrite (lt_le_trans (m1M (ysection X y) _))// ?leey//. exact: measurable_ysection. exact: emeasurable_funB. Qed. End ysection. End measurable_prod_subset. Section measurable_fun_xsection. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Variables (m2 : {measure set T2 -> \bar R}). Hypothesis sf_m2 : sigma_finite setT m2. Implicit Types A : set (T1 * T2). Let phi A := m2 \o xsection A. Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. Lemma measurable_fun_xsection A : A \in measurable -> measurable_fun setT (phi A). Proof. move: A; suff : measurable `<=` B by move=> + A; rewrite inE => /[apply] -[]. move/sigma_finiteP : sf_m2 => [F F_T [F_nd F_oo]] X mX. have -> : X = \bigcup_n (X `&` (setT `*` F n)). by rewrite -setI_bigcupr -setM_bigcupr -F_T setMTT setIT. apply: xsection_ndseq_closed. move=> m n mn; apply/subsetPset; apply: setIS; apply: setSM => //. exact/subsetPset/F_nd. move=> n; rewrite -/B; have [? ?] := F_oo n. pose m2Fn := [the measure _ _ of mrestr m2 (F_oo n).1]. have m2Fn_bounded : exists M, forall X, measurable X -> (m2Fn X < M%:E)%E. exists (fine (m2Fn (F n)) + 1) => Y mY. rewrite [in ltRHS]EFinD (le_lt_trans _ (lte_addl _ _)) ?lte_fin//. rewrite fineK; last first. by rewrite ge0_fin_numE ?measure_ge0//= /mrestr/= setIid. rewrite /= /mrestr/= setIid; apply: le_measure => //; rewrite inE//. exact: measurableI. pose phi' A := m2Fn \o xsection A. pose B' := [set A | measurable A /\ measurable_fun setT (phi' A)]. have subset_B' : measurable `<=` B' by exact: measurable_prod_subset_xsection. split=> [|Y mY]; first by apply: measurableI => //; exact: measurableM. have [_ /(_ Y mY)] := subset_B' X mX. have ->// : phi' X = (fun x => m2 [set y | (x, y) \in X `&` setT `*` F n]). rewrite funeqE => x /=; congr (m2 _); rewrite predeqE => y; split => [[]|]. by rewrite /xsection /= inE => Xxy Fny; rewrite inE. by rewrite /xsection /= !inE => -[] Xxy /= [_]. Qed. End measurable_fun_xsection. Section measurable_fun_ysection. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Variables (m1 : {measure set T1 -> \bar R}). Hypothesis sf_m1 : sigma_finite setT m1. Implicit Types A : set (T1 * T2). Let phi A := m1 \o ysection A. Let B := [set A | measurable A /\ measurable_fun setT (phi A)]. Lemma measurable_fun_ysection A : A \in measurable -> measurable_fun setT (phi A). Proof. move: A; suff : measurable `<=` B by move=> + A; rewrite inE => /[apply] -[]. move : sf_m1 => /sigma_finiteP[F F_T [F_nd F_oo]] X mX. have -> : X = \bigcup_n (X `&` (F n `*` setT)). by rewrite -setI_bigcupr -setM_bigcupl -F_T setMTT setIT. apply: ysection_ndseq_closed. move=> m n mn; apply/subsetPset; apply: setIS; apply: setSM => //. exact/subsetPset/F_nd. move=> n; have [? ?] := F_oo n; rewrite -/B. pose m1Fn := [the measure _ _ of mrestr m1 (F_oo n).1]. have m1Fn_bounded : exists M, forall X, measurable X -> (m1Fn X < M%:E)%E. exists (fine (m1Fn (F n)) + 1) => Y mY. rewrite [in ltRHS]EFinD (le_lt_trans _ (lte_addl _ _)) ?lte_fin//. rewrite fineK; last first. by rewrite ge0_fin_numE ?measure_ge0// /m1Fn/= /mrestr setIid. rewrite /m1Fn/= /mrestr setIid; apply: le_measure => //; rewrite inE//=. exact: measurableI. pose psi' A := m1Fn \o ysection A. pose B' := [set A | measurable A /\ measurable_fun setT (psi' A)]. have subset_B' : measurable `<=` B'. exact: measurable_prod_subset_ysection. split=> [|Y mY]; first by apply: measurableI => //; exact: measurableM. have [_ /(_ Y mY)] := subset_B' X mX. have ->// : psi' X = (fun y => m1 [set x | (x, y) \in X `&` F n `*` setT]). rewrite funeqE => y /=; congr (m1 _); rewrite predeqE => x; split => [[]|]. by rewrite /ysection /= inE => Xxy Fny; rewrite inE. by rewrite /ysection /= !inE => -[] Xxy/= []. Qed. End measurable_fun_ysection. Definition product_measure1 d1 d2 (T1 : measurableType d1) (T2 : measurableType d2) (R : realType) (m1 : {measure set T1 -> \bar R}) (m2 : {measure set T2 -> \bar R}) (sm2 : sigma_finite setT m2) := (fun A => \int[m1]_x (m2 \o xsection A) x)%E. Section product_measure1. Local Open Scope ereal_scope. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Variables (m1 : {measure set T1 -> \bar R}) (m2 : {measure set T2 -> \bar R}). Hypothesis sm2 : sigma_finite setT m2. Implicit Types A : set (T1 * T2). Notation pm1 := (product_measure1 m1 sm2). Let pm10 : pm1 set0 = 0. Proof. rewrite /pm1 (eq_integral (cst 0)) ?integral0//= => x _. by rewrite xsection0 measure0. Qed. Let pm1_ge0 A : 0 <= pm1 A. Proof. by apply: integral_ge0 => // *; exact/measure_ge0/measurable_xsection. Qed. Let pm1_sigma_additive : semi_sigma_additive pm1. Proof. move=> F mF tF mUF; have -> : pm1 (\bigcup_n F n) = \sum_(n x _; apply/esym/cvg_lim => //=. rewrite xsection_bigcup. apply: (measure_sigma_additive _ (trivIset_xsection tF)). by move=> ?; exact: measurable_xsection. by rewrite integral_sum // => n; apply: measurable_fun_xsection => // /[!inE]. apply/cvg_closeP; split; last by rewrite closeE. by apply: is_cvg_nneseries => *; exact: integral_ge0. Qed. HB.instance Definition _ := isMeasure.Build _ _ _ pm1 pm10 pm1_ge0 pm1_sigma_additive. End product_measure1. Section product_measure1E. Local Open Scope ereal_scope. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Variables (m1 : {measure set T1 -> \bar R}) (m2 : {measure set T2 -> \bar R}). Hypothesis sm2 : sigma_finite setT m2. Implicit Types A : set (T1 * T2). Lemma product_measure1E (A1 : set T1) (A2 : set T2) : measurable A1 -> measurable A2 -> product_measure1 m1 sm2 (A1 `*` A2) = m1 A1 * m2 A2. Proof. move=> mA1 mA2 /=; rewrite /product_measure1 /=. rewrite (_ : (fun _ => _) = fun x => m2 A2 * (\1_A1 x)%:E); last first. rewrite funeqE => x; rewrite indicE. by have [xA1|xA1] /= := boolP (x \in A1); [rewrite in_xsectionM// mule1|rewrite mule0 notin_xsectionM]. rewrite ge0_integralM//. - by rewrite muleC integral_indic// setIT. - by apply: measurable_fun_comp => //; rewrite (_ : \1_ _ = mindic R mA1). - by move=> x _; rewrite lee_fin. Qed. End product_measure1E. Section product_measure_unique. Local Open Scope ereal_scope. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Variables (m1 : {measure set T1 -> \bar R}) (m2 : {measure set T2 -> \bar R}). Hypotheses (sm1 : sigma_finite setT m1) (sm2 : sigma_finite setT m2). Lemma product_measure_unique (m' : {measure set [the semiRingOfSetsType _ of (T1 * T2)%type] -> \bar R}) : (forall A1 A2, measurable A1 -> measurable A2 -> m' (A1 `*` A2) = m1 A1 * m2 A2) -> forall X : set (T1 * T2), measurable X -> product_measure1 m1 sm2 X = m' X. Proof. move=> m'E; pose m := product_measure1 m1 sm2. move/sigma_finiteP : sm1 => [F1 F1_T [F1_nd F1_oo]]. move/sigma_finiteP : sm2 => [F2 F2_T [F2_nd F2_oo]]. have UF12T : \bigcup_k (F1 k `*` F2 k) = setT. rewrite -setMTT F1_T F2_T predeqE => -[x y]; split. by move=> [n _ []/= ? ?]; split; exists n. move=> [/= [n _ F1nx] [k _ F2ky]]; exists (maxn n k) => //; split. - by move: x F1nx; apply/subsetPset/F1_nd; rewrite leq_maxl. - by move: y F2ky; apply/subsetPset/F2_nd; rewrite leq_maxr. have mF1F2 n : measurable (F1 n `*` F2 n) /\ m (F1 n `*` F2 n) < +oo. have [? ?] := F1_oo n; have [? ?] := F2_oo n. split; first exact: measurableM. by rewrite /m product_measure1E // lte_mul_pinfty// ge0_fin_numE. have sm : sigma_finite setT m by exists (fun n => F1 n `*` F2 n). pose C : set (set (T1 * T2)) := [set C | exists A1, measurable A1 /\ exists A2, measurable A2 /\ C = A1 `*` A2]. have CI : setI_closed C. move=> /= _ _ [X1 [mX1 [X2 [mX2 ->]]]] [Y1 [mY1 [Y2 [mY2 ->]]]]. rewrite -setMI; exists (X1 `&` Y1); split; first exact: measurableI. by exists (X2 `&` Y2); split => //; exact: measurableI. move=> X mX; apply: (measure_unique C (fun n => F1 n `*` F2 n)) => //. - rewrite measurable_prod_measurableType //; congr (<>). rewrite predeqE; split => [[A1 mA1 [A2 mA2 <-]]|[A1 [mA1 [A2 [mA2 ->]]]]]. by exists A1; split => //; exists A2; split. by exists A1 => //; exists A2. - move=> n; rewrite /C /=; exists (F1 n); split; first by have [] := F1_oo n. by exists (F2 n); split => //; have [] := F2_oo n. - by move=> A [A1 [mA1 [A2 [mA2 ->]]]]; rewrite m'E//= product_measure1E. - move=> k; have [? ?] := F1_oo k; have [? ?] := F2_oo k. by rewrite /= product_measure1E// lte_mul_pinfty// ge0_fin_numE. Qed. End product_measure_unique. Definition product_measure2 d1 d2 (T1 : measurableType d1) (T2 : measurableType d2) (R : realType) (m1 : {measure set T1 -> \bar R}) (sm1 : sigma_finite setT m1) (m2 : {measure set T2 -> \bar R}) := (fun A => \int[m2]_x (m1 \o ysection A) x)%E. Section product_measure2. Local Open Scope ereal_scope. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Variables (m1 : {measure set T1 -> \bar R}) (m2 : {measure set T2 -> \bar R}). Hypothesis sm1 : sigma_finite setT m1. Implicit Types A : set (T1 * T2). Notation pm2 := (product_measure2 sm1 m2). Let pm20 : pm2 set0 = 0. Proof. rewrite /pm2 (eq_integral (cst 0)) ?integral0//= => y _. by rewrite ysection0 measure0. Qed. Let pm2_ge0 A : 0 <= pm2 A. Proof. by apply: integral_ge0 => // *; exact/measure_ge0/measurable_ysection. Qed. Let pm2_sigma_additive : semi_sigma_additive pm2. Proof. move=> F mF tF mUF. have -> : pm2 (\bigcup_n F n) = \sum_(n y _; apply/esym/cvg_lim => //=. rewrite ysection_bigcup. apply: (measure_sigma_additive _ (trivIset_ysection tF)). by move=> ?; apply: measurable_ysection. by rewrite integral_sum // => n; apply: measurable_fun_ysection => // /[!inE]. apply/cvg_closeP; split; last by rewrite closeE. by apply: is_cvg_nneseries => *; exact: integral_ge0. Qed. HB.instance Definition _ := isMeasure.Build _ _ _ pm2 pm20 pm2_ge0 pm2_sigma_additive. End product_measure2. Section product_measure2E. Local Open Scope ereal_scope. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Variables (m1 : {measure set T1 -> \bar R}) (m2 : {measure set T2 -> \bar R}). Hypothesis sm1 : sigma_finite setT m1. Lemma product_measure2E (A1 : set T1) (A2 : set T2) (mA1 : measurable A1) (mA2 : measurable A2) : product_measure2 sm1 m2 (A1 `*` A2) = m1 A1 * m2 A2. Proof. have mA1A2 : measurable (A1 `*` A2) by apply: measurableM. transitivity (\int[m2]_y (m1 \o ysection (A1 `*` A2)) y) => //. rewrite (_ : _ \o _ = fun y => m1 A1 * (\1_A2 y)%:E). rewrite ge0_integralM//; last 2 first. - by apply: measurable_fun_comp => //; rewrite (_ : \1_ _ = mindic R mA2). - by move=> y _; rewrite lee_fin. by rewrite integral_indic ?setIT ?mul1e. rewrite funeqE => y; rewrite indicE. have [yA2|yA2] := boolP (y \in A2); first by rewrite mule1 /= in_ysectionM. by rewrite mule0 /= notin_ysectionM// measure0. Qed. End product_measure2E. Section fubini_functions. Local Open Scope ereal_scope. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Variables (m1 : {measure set T1 -> \bar R}) (m2 : {measure set T2 -> \bar R}). Variable f : T1 * T2 -> \bar R. Definition fubini_F x := \int[m2]_y f (x, y). Definition fubini_G y := \int[m1]_x f (x, y). End fubini_functions. Section fubini_tonelli. Local Open Scope ereal_scope. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Variables (m1 : {measure set T1 -> \bar R}) (m2 : {measure set T2 -> \bar R}). Hypotheses (sm1 : sigma_finite setT m1) (sm2 : sigma_finite setT m2). Let m := product_measure1 m1 sm2. Let m' := product_measure2 sm1 m2. HB.instance Definition _ := Measure.on m. HB.instance Definition _ := Measure.on m'. Section indic_fubini_tonelli. Variables (A : set (T1 * T2)) (mA : measurable A). Implicit Types A : set (T1 * T2). Let f : (T1 * T2) -> R := \1_A. Let F := fubini_F m2 (EFin \o f). Let G := fubini_G m1 (EFin \o f). Lemma indic_fubini_tonelli_F_ge0 x : 0 <= F x. Proof. by apply: integral_ge0 => // y _; rewrite lee_fin. Qed. Lemma indic_fubini_tonelli_G_ge0 x : 0 <= G x. Proof. by apply: integral_ge0 => // y _; rewrite lee_fin. Qed. Lemma indic_fubini_tonelli_FE : F = m2 \o xsection A. Proof. rewrite funeqE => x; rewrite /= -(setTI (xsection _ _)). rewrite -integral_indic//; last exact: measurable_xsection. rewrite /F /fubini_F -(setTI (xsection _ _)). rewrite integral_setI_indic; [|exact: measurable_xsection|by []]. apply: eq_integral => y _ /=; rewrite indicT mul1e /f !indicE. have [|] /= := boolP (y \in xsection _ _). by rewrite inE /xsection /= => ->. by rewrite /xsection /= notin_set /= => /negP/negbTE ->. Qed. Lemma indic_fubini_tonelli_GE : G = m1 \o ysection A. Proof. rewrite funeqE => y; rewrite /= -(setTI (ysection _ _)). rewrite -integral_indic//; last exact: measurable_ysection. rewrite /F /fubini_F -(setTI (ysection _ _)). rewrite integral_setI_indic; [|exact: measurable_ysection|by []]. apply: eq_integral => x _ /=; rewrite indicT mul1e /f 2!indicE. have [|] /= := boolP (x \in ysection _ _). by rewrite inE /xsection /= => ->. by rewrite /xsection /= notin_set /= => /negP/negbTE ->. Qed. Lemma indic_measurable_fun_fubini_tonelli_F : measurable_fun setT F. Proof. rewrite indic_fubini_tonelli_FE//; apply: measurable_fun_xsection => //. by rewrite inE. Qed. Lemma indic_measurable_fun_fubini_tonelli_G : measurable_fun setT G. Proof. rewrite indic_fubini_tonelli_GE//; apply: measurable_fun_ysection => //. by rewrite inE. Qed. Let mE : m A = \int[m1]_x F x. Proof. by rewrite /m /product_measure1 /= indic_fubini_tonelli_FE. Qed. Lemma indic_fubini_tonelli1 : \int[m]_z (f z)%:E = \int[m1]_x F x. Proof. by rewrite /f integral_indic// setIT indic_fubini_tonelli_FE. Qed. Lemma indic_fubini_tonelli2 : \int[m']_z (f z)%:E = \int[m2]_y G y. by rewrite /f integral_indic// setIT indic_fubini_tonelli_GE. Qed. Lemma indic_fubini_tonelli : \int[m1]_x F x = \int[m2]_y G y. Proof. rewrite -indic_fubini_tonelli1// -indic_fubini_tonelli2//. rewrite integral_indic // integral_indic // setIT/=. by apply: product_measure_unique => //= ? ? ? ?; rewrite /m' product_measure2E. Qed. End indic_fubini_tonelli. Section sfun_fubini_tonelli. Variable f : {nnsfun [the measurableType _ of T1 * T2 : Type] >-> R}. Let F := fubini_F m2 (EFin \o f). Let G := fubini_G m1 (EFin \o f). Lemma sfun_fubini_tonelli_FE : F = fun x => \sum_(k <- fset_set (range f)) k%:E * m2 (xsection (f @^-1` [set k]) x). Proof. rewrite funeqE => x; rewrite /F /fubini_F [in LHS]/=. under eq_fun do rewrite fimfunE -sumEFin. rewrite ge0_integral_sum //; last 2 first. - move=> i; apply/EFin_measurable_fun => //; apply: measurable_funrM => //. apply/measurable_fun_prod1 => //. (*NB: we shouldn't need the following rewriting*) by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f i)). - by move=> r y _; rewrite EFinM; exact: muleindic_ge0. apply: eq_fbigr => i; rewrite in_fset_set// inE => -[/= t _ <-{i} _]. under eq_fun do rewrite EFinM. rewrite ge0_integralM//; last by rewrite lee_fin. - by rewrite -/((m2 \o xsection _) x) -indic_fubini_tonelli_FE. - apply/EFin_measurable_fun/measurable_fun_prod1. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f (f t))). - by move=> y _; rewrite lee_fin. Qed. Lemma sfun_measurable_fun_fubini_tonelli_F : measurable_fun setT F. Proof. rewrite sfun_fubini_tonelli_FE//; apply: emeasurable_fun_sum => // r. by apply: emeasurable_funeM => //; apply: measurable_fun_xsection => // /[!inE]. Qed. Lemma sfun_fubini_tonelli_GE : G = fun y => \sum_(k <- fset_set (range f)) k%:E * m1 (ysection (f @^-1` [set k]) y). Proof. rewrite funeqE => y; rewrite /G /fubini_G [in LHS]/=. under eq_fun do rewrite fimfunE -sumEFin. rewrite ge0_integral_sum //; last 2 first. - move=> i; apply/EFin_measurable_fun => //; apply: measurable_funrM => //. apply/measurable_fun_prod2 => //. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f i)). - by move=> r x _; rewrite EFinM; exact: muleindic_ge0. apply: eq_fbigr => i; rewrite in_fset_set// inE => -[/= t _ <-{i} _]. under eq_fun do rewrite EFinM. rewrite ge0_integralM//; last by rewrite lee_fin. - by rewrite -/((m1 \o ysection _) y) -indic_fubini_tonelli_GE. - apply/EFin_measurable_fun/measurable_fun_prod2. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f (f t))). - by move=> x _; rewrite lee_fin. Qed. Lemma sfun_measurable_fun_fubini_tonelli_G : measurable_fun setT G. Proof. rewrite sfun_fubini_tonelli_GE//; apply: emeasurable_fun_sum => // r. by apply: emeasurable_funeM => //; apply: measurable_fun_ysection => // /[!inE]. Qed. Let EFinf x : (f x)%:E = \sum_(k <- fset_set (range f)) k%:E * (\1_(f @^-1` [set k]) x)%:E. Proof. by rewrite sumEFin /= fimfunE. Qed. Lemma sfun_fubini_tonelli1 : \int[m]_z (f z)%:E = \int[m1]_x F x. Proof. under [LHS]eq_integral do rewrite EFinf; rewrite ge0_integral_sum //; last 2 first. - move=> r; apply/EFin_measurable_fun/measurable_funrM => //. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f r)). - by move=> r /= z _; exact: muleindic_ge0. transitivity (\sum_(k <- fset_set (range f)) \int[m1]_x (k%:E * (fubini_F m2 (EFin \o \1_(f @^-1` [set k])) x))). apply: eq_fbigr => i; rewrite in_fset_set// inE => -[z _ <-{i} _]. rewrite ge0_integralM//; last 3 first. - apply/EFin_measurable_fun. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f (f z)))//. - by move=> /= x _; rewrite lee_fin. - by rewrite lee_fin. rewrite indic_fubini_tonelli1// -ge0_integralM//; last by rewrite lee_fin. - exact: indic_measurable_fun_fubini_tonelli_F. - by move=> /= x _; exact: indic_fubini_tonelli_F_ge0. rewrite -ge0_integral_sum //; last 2 first. - move=> /= r; apply: emeasurable_funeM => //. exact: indic_measurable_fun_fubini_tonelli_F. - move=> r x _; rewrite /fubini_F. have [r0|r0] := leP 0%R r. by rewrite mule_ge0//; exact: indic_fubini_tonelli_F_ge0. rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0// => y _. by rewrite preimage_nnfun0//= indicE in_set0. apply: eq_integral => x _; rewrite sfun_fubini_tonelli_FE. by apply: eq_bigr => ? _; rewrite indic_fubini_tonelli_FE. Qed. Lemma sfun_fubini_tonelli2 : \int[m']_z (f z)%:E = \int[m2]_y G y. Proof. under [LHS]eq_integral do rewrite EFinf; rewrite ge0_integral_sum //; last 2 first. - move=> i; apply/EFin_measurable_fun/measurable_funrM => //. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f i)). - by move=> r /= z _; exact: muleindic_ge0. transitivity (\sum_(k <- fset_set (range f)) \int[m2]_x (k%:E * (fubini_G m1 (EFin \o \1_(f @^-1` [set k])) x))). apply: eq_fbigr => i; rewrite in_fset_set// inE => -[z _ <-{i} _]. rewrite ge0_integralM//; last 3 first. - apply/EFin_measurable_fun. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f (f z))). - by move=> /= x _; rewrite lee_fin. - by rewrite lee_fin. rewrite indic_fubini_tonelli2// -ge0_integralM//; last by rewrite lee_fin. - exact: indic_measurable_fun_fubini_tonelli_G. - by move=> /= x _; exact: indic_fubini_tonelli_G_ge0. rewrite -ge0_integral_sum //; last 2 first. - move=> /= i; apply: emeasurable_funeM => //. exact: indic_measurable_fun_fubini_tonelli_G. - move=> r x _; rewrite /fubini_G. have [r0|r0] := leP 0%R r. by rewrite mule_ge0//; exact: indic_fubini_tonelli_G_ge0. rewrite (eq_integral (cst 0%E)) ?integral0 ?mule0// => y _. by rewrite preimage_nnfun0//= indicE in_set0. apply: eq_integral => x _; rewrite sfun_fubini_tonelli_GE. by apply: eq_bigr => i _; rewrite indic_fubini_tonelli_GE. Qed. Lemma sfun_fubini_tonelli : \int[m]_z (f z)%:E = \int[m']_z (f z)%:E. Proof. under eq_integral do rewrite EFinf. under [RHS]eq_integral do rewrite EFinf. rewrite ge0_integral_sum //; last 2 first. - move=> i; apply/EFin_measurable_fun/measurable_funrM => //. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f i)). - by move=> r z _; exact: muleindic_ge0. transitivity (\sum_(k <- fset_set (range f)) k%:E * \int[m']_z ((EFin \o \1_(f @^-1` [set k])) z)). apply: eq_fbigr => i; rewrite in_fset_set// inE => -[t _ <- _]. rewrite ge0_integralM//; last 3 first. - apply/EFin_measurable_fun. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f (f t))). - by move=> /= x _; rewrite lee_fin. - by rewrite lee_fin. rewrite indic_fubini_tonelli1// indic_fubini_tonelli//. by rewrite -indic_fubini_tonelli2. apply/esym; rewrite ge0_integral_sum //; last 2 first. - move=> i; apply/EFin_measurable_fun/measurable_funrM => //. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f i)). - by move=> r z _; exact: muleindic_ge0. apply: eq_fbigr => i; rewrite in_fset_set// inE => -[x _ <- _]. rewrite ge0_integralM//; last by rewrite lee_fin. - apply/EFin_measurable_fun. by rewrite (_ : \1_ _ = mindic R (measurable_sfunP f (f x))). - by move=> /= y _; rewrite lee_fin. Qed. End sfun_fubini_tonelli. Section fubini_tonelli. Variable f : T1 * T2 -> \bar R. Hypothesis mf : measurable_fun setT f. Hypothesis f0 : forall x, 0 <= f x. Let T := [the measurableType _ of T1 * T2 : Type]. Let F := fubini_F m2 f. Let G := fubini_G m1 f. Let F_ (g : {nnsfun T >-> R}^nat) n x := \int[m2]_y (g n (x, y))%:E. Let G_ (g : {nnsfun T >-> R}^nat) n y := \int[m1]_x (g n (x, y))%:E. Lemma measurable_fun_fubini_tonelli_F : measurable_fun setT F. Proof. have [g [g_nd /= g_f]] := approximation measurableT mf (fun x _ => f0 x). apply: (emeasurable_fun_cvg (F_ g)) => //. - by move=> n; exact: sfun_measurable_fun_fubini_tonelli_F. - move=> x _. rewrite /F_ /F /fubini_F [in X in _ --> X](_ : (fun _ => _) = fun y => lim (EFin \o g ^~ (x, y))); last first. by rewrite funeqE => y; apply/esym/cvg_lim => //; exact: g_f. apply: cvg_monotone_convergence => //. - by move=> n; apply/EFin_measurable_fun => //; exact/measurable_fun_prod1. - by move=> n y _; rewrite lee_fin//; exact: fun_ge0. - by move=> y _ a b ab; rewrite lee_fin; exact/lefP/g_nd. Qed. Lemma measurable_fun_fubini_tonelli_G : measurable_fun setT G. Proof. have [g [g_nd /= g_f]] := approximation measurableT mf (fun x _ => f0 x). apply: (emeasurable_fun_cvg (G_ g)) => //. - by move=> n; exact: sfun_measurable_fun_fubini_tonelli_G. - move=> y _; rewrite /G_ /G /fubini_G [in X in _ --> X](_ : (fun _ => _) = fun x => lim (EFin \o g ^~ (x, y))); last first. by rewrite funeqE => x; apply/esym/cvg_lim => //; exact: g_f. apply: cvg_monotone_convergence => //. - by move=> n; apply/EFin_measurable_fun => //; exact/measurable_fun_prod2. - by move=> n x _; rewrite lee_fin; exact: fun_ge0. - by move=> x _ a b ab; rewrite lee_fin; exact/lefP/g_nd. Qed. Lemma fubini_tonelli1 : \int[m]_z f z = \int[m1]_x F x. Proof. have [g [g_nd /= g_f]] := approximation measurableT mf (fun x _ => f0 x). have F_F x : F x = lim (F_ g ^~ x). rewrite /F /fubini_F. rewrite [RHS](_ : _ = lim (fun n => \int[m2]_y (EFin \o g n) (x, y)))//. rewrite -monotone_convergence//; last 3 first. - by move=> n; exact/EFin_measurable_fun/measurable_fun_prod1. - by move=> n /= y _; rewrite lee_fin; exact: fun_ge0. - by move=> y /= _ a b; rewrite lee_fin => /g_nd/lefP; exact. by apply: eq_integral => y _; apply/esym/cvg_lim => //; exact: g_f. rewrite [RHS](_ : _ = lim (fun n => \int[m]_z (EFin \o g n) z)). rewrite -monotone_convergence //; last 3 first. - by move=> n; exact/EFin_measurable_fun. - by move=> n /= x _; rewrite lee_fin; exact: fun_ge0. - by move=> y /= _ a b; rewrite lee_fin => /g_nd/lefP; exact. by apply: eq_integral => /= x _; apply/esym/cvg_lim => //; exact: g_f. rewrite [LHS](_ : _ = lim (fun n => \int[m1]_x (\int[m2]_y (EFin \o g n) (x, y)))). by congr (lim _); rewrite funeqE => n; rewrite sfun_fubini_tonelli1. rewrite [RHS](_ : _ = lim (fun n => \int[m1]_x F_ g n x))//. rewrite -monotone_convergence //; first exact: eq_integral. - by move=> n; exact: sfun_measurable_fun_fubini_tonelli_F. - move=> n x _; apply: integral_ge0 => // y _ /=; rewrite lee_fin. exact: fun_ge0. - move=> x /= _ a b ab; apply: ge0_le_integral => //. + by move=> y _; rewrite lee_fin; exact: fun_ge0. + exact/EFin_measurable_fun/measurable_fun_prod1. + by move=> *; rewrite lee_fin; exact: fun_ge0. + exact/EFin_measurable_fun/measurable_fun_prod1. + by move=> y _; rewrite lee_fin; move/g_nd : ab => /lefP; exact. Qed. Lemma fubini_tonelli2 : \int[m]_z f z = \int[m2]_y G y. Proof. have [g [g_nd /= g_f]] := approximation measurableT mf (fun x _ => f0 x). have G_G y : G y = lim (G_ g ^~ y). rewrite /G /fubini_G. rewrite [RHS](_ : _ = lim (fun n => \int[m1]_x (EFin \o g n) (x, y)))//. rewrite -monotone_convergence//; last 3 first. - by move=> n; exact/EFin_measurable_fun/measurable_fun_prod2. - by move=> n /= x _; rewrite lee_fin; exact: fun_ge0. - by move=> x /= _ a b; rewrite lee_fin => /g_nd/lefP; exact. by apply: eq_integral => x _; apply/esym/cvg_lim => //; exact: g_f. rewrite [RHS](_ : _ = lim (fun n => \int[m]_z (EFin \o g n) z)). rewrite -monotone_convergence //; last 3 first. - by move=> n; exact/EFin_measurable_fun. - by move=> n /= x _; rewrite lee_fin; exact: fun_ge0. - by move=> y /= _ a b; rewrite lee_fin => /g_nd/lefP; exact. by apply: eq_integral => /= x _; apply/esym/cvg_lim => //; exact: g_f. rewrite [LHS](_ : _ = lim (fun n => \int[m2]_y (\int[m1]_x (EFin \o g n) (x, y)))). congr (lim _); rewrite funeqE => n. by rewrite sfun_fubini_tonelli sfun_fubini_tonelli2. rewrite [RHS](_ : _ = lim (fun n => \int[m2]_y G_ g n y))//. rewrite -monotone_convergence //; first exact: eq_integral. - by move=> n; exact: sfun_measurable_fun_fubini_tonelli_G. - by move=> n y _; apply: integral_ge0 => // x _ /=; rewrite lee_fin fun_ge0. - move=> y /= _ a b ab; apply: ge0_le_integral => //. + by move=> x _; rewrite lee_fin fun_ge0. + exact/EFin_measurable_fun/measurable_fun_prod2. + by move=> *; rewrite lee_fin fun_ge0. + exact/EFin_measurable_fun/measurable_fun_prod2. + by move=> x _; rewrite lee_fin; move/g_nd : ab => /lefP; exact. Qed. End fubini_tonelli. End fubini_tonelli. Arguments fubini_tonelli1 {d1 d2 T1 T2 R m1 m2} sm2 f. Arguments fubini_tonelli2 {d1 d2 T1 T2 R m1 m2} sm1 sm2 f. Arguments measurable_fun_fubini_tonelli_F {d1 d2 T1 T2 R m2} sm2 f. Arguments measurable_fun_fubini_tonelli_G {d1 d2 T1 T2 R m1} sm1 f. Section fubini. Local Open Scope ereal_scope. Variables (d1 d2 : measure_display). Variables (T1 : measurableType d1) (T2 : measurableType d2) (R : realType). Variables (m1 : {measure set T1 -> \bar R}) (m2 : {measure set T2 -> \bar R}). Hypotheses (sm1 : sigma_finite setT m1) (sm2 : sigma_finite setT m2). Variable f : T1 * T2 -> \bar R. Hypothesis mf : measurable_fun setT f. Let m := product_measure1 m1 sm2. HB.instance Definition _ := Measure.on m. Lemma fubini1a : m.-integrable setT f <-> \int[m1]_x \int[m2]_y `|f (x, y)| < +oo. Proof. split=> [[_]|] ioo. - by rewrite -(fubini_tonelli1 _ (abse \o f))//=; exact: measurable_fun_comp. - by split=> //; rewrite fubini_tonelli1//; exact: measurable_fun_comp. Qed. Lemma fubini1b : m.-integrable setT f <-> \int[m2]_y \int[m1]_x `|f (x, y)| < +oo. Proof. split=> [[_]|] ioo. - by rewrite -(fubini_tonelli2 _ _ (abse \o f))//=; exact: measurable_fun_comp. - by split=> //; rewrite fubini_tonelli2//; exact: measurable_fun_comp. Qed. Let measurable_fun1 : measurable_fun setT (fun x => \int[m2]_y `|f (x, y)|). Proof. apply: (measurable_fun_fubini_tonelli_F _ (abse \o f)) => //=. exact: measurable_fun_comp. Qed. Let measurable_fun2 : measurable_fun setT (fun y => \int[m1]_x `|f (x, y)|). Proof. apply: (measurable_fun_fubini_tonelli_G _ (abse \o f)) => //=. exact: measurable_fun_comp. Qed. Hypothesis imf : m.-integrable setT f. Lemma ae_integrable1 : {ae m1, forall x, m2.-integrable setT (fun y => f (x, y))}. Proof. have : m1.-integrable setT (fun x => \int[m2]_y `|f (x, y)|). split => //; rewrite (le_lt_trans _ (fubini1a.1 imf))// ge0_le_integral //. - exact: measurable_fun_comp. - by move=> *; exact: integral_ge0. - by move=> *; rewrite gee0_abs//; exact: integral_ge0. move/integrable_ae => /(_ measurableT) [N [mN N0 subN]]; exists N; split => //. apply/(subset_trans _ subN)/subsetC => x /= /(_ Logic.I) im2f. by split; [exact/measurable_fun_prod1|by move/fin_numPlt : im2f => /andP[]]. Qed. Lemma ae_integrable2 : {ae m2, forall y, m1.-integrable setT (fun x => f (x, y))}. Proof. have : m2.-integrable setT (fun y => \int[m1]_x `|f (x, y)|). split => //; rewrite (le_lt_trans _ (fubini1b.1 imf))// ge0_le_integral //. - exact: measurable_fun_comp. - by move=> *; exact: integral_ge0. - by move=> *; rewrite gee0_abs//; exact: integral_ge0. move/integrable_ae => /(_ measurableT) [N [mN N0 subN]]; exists N; split => //. apply/(subset_trans _ subN)/subsetC => x /= /(_ Logic.I) im1f. by split; [exact/measurable_fun_prod2|move/fin_numPlt : im1f => /andP[]]. Qed. Let F := fubini_F m2 f. Let Fplus x := \int[m2]_y f^\+ (x, y). Let Fminus x := \int[m2]_y f^\- (x, y). Let FE : F = Fplus \- Fminus. Proof. apply/funext=> x; exact: integralE. Qed. Let measurable_Fplus : measurable_fun setT Fplus. Proof. by apply: measurable_fun_fubini_tonelli_F => //; exact: emeasurable_fun_funepos. Qed. Let measurable_Fminus : measurable_fun setT Fminus. Proof. by apply: measurable_fun_fubini_tonelli_F => //; exact: emeasurable_fun_funeneg. Qed. Lemma measurable_fubini_F : measurable_fun setT F. Proof. rewrite FE. by apply: emeasurable_funB; [exact: measurable_Fplus|exact: measurable_Fminus]. Qed. Let integrable_Fplus : m1.-integrable setT Fplus. Proof. split=> //; apply: le_lt_trans (fubini1a.1 imf); apply: ge0_le_integral => //. - exact: measurable_fun_comp. - by move=> x _; exact: integral_ge0. - move=> x _; apply: le_trans. apply: le_abse_integral => //; apply: emeasurable_fun_funepos => //. exact: measurable_fun_prod1. apply: ge0_le_integral => //. - apply: measurable_fun_comp => //. by apply: emeasurable_fun_funepos => //; exact: measurable_fun_prod1. - by apply: measurable_fun_comp => //; exact/measurable_fun_prod1. - by move=> y _; rewrite gee0_abs// -/((abse \o f) (x, y)) fune_abse lee_addl. Qed. Let integrable_Fminus : m1.-integrable setT Fminus. Proof. split=> //; apply: le_lt_trans (fubini1a.1 imf); apply: ge0_le_integral => //. - exact: measurable_fun_comp. - by move=> *; exact: integral_ge0. - move=> x _; apply: le_trans. apply: le_abse_integral => //; apply: emeasurable_fun_funeneg => //. exact: measurable_fun_prod1. apply: ge0_le_integral => //. + apply: measurable_fun_comp => //; apply: emeasurable_fun_funeneg => //. exact: measurable_fun_prod1. + by apply: measurable_fun_comp => //; exact: measurable_fun_prod1. + by move=> y _; rewrite gee0_abs// -/((abse \o f) (x, y)) fune_abse lee_addr. Qed. Lemma integrable_fubini_F : m1.-integrable setT F. Proof. by rewrite FE; exact: integrableB. Qed. Let G := fubini_G m1 f. Let Gplus y := \int[m1]_x f^\+ (x, y). Let Gminus y := \int[m1]_x f^\- (x, y). Let GE : G = Gplus \- Gminus. Proof. apply/funext=> x; exact: integralE. Qed. Let measurable_Gplus : measurable_fun setT Gplus. Proof. by apply: measurable_fun_fubini_tonelli_G => //; exact: emeasurable_fun_funepos. Qed. Let measurable_Gminus : measurable_fun setT Gminus. Proof. by apply: measurable_fun_fubini_tonelli_G => //; exact: emeasurable_fun_funeneg. Qed. Lemma measurable_fubini_G : measurable_fun setT G. Proof. by rewrite GE; exact: emeasurable_funB. Qed. Let integrable_Gplus : m2.-integrable setT Gplus. Proof. split=> //; apply: le_lt_trans (fubini1b.1 imf); apply: ge0_le_integral => //. - exact: measurable_fun_comp. - by move=> *; exact: integral_ge0. - move=> y _; apply: le_trans. apply: le_abse_integral => //; apply: emeasurable_fun_funepos => //. exact: measurable_fun_prod2. apply: ge0_le_integral => //. - apply: measurable_fun_comp => //. by apply: emeasurable_fun_funepos => //; exact: measurable_fun_prod2. - by apply: measurable_fun_comp => //; exact: measurable_fun_prod2. - by move=> x _; rewrite gee0_abs// -/((abse \o f) (x, y)) fune_abse lee_addl. Qed. Let integrable_Gminus : m2.-integrable setT Gminus. Proof. split=> //; apply: le_lt_trans (fubini1b.1 imf); apply: ge0_le_integral => //. - exact: measurable_fun_comp. - by move=> *; exact: integral_ge0. - move=> y _; apply: le_trans. apply: le_abse_integral => //; apply: emeasurable_fun_funeneg => //. exact: measurable_fun_prod2. apply: ge0_le_integral => //. + apply: measurable_fun_comp => //. by apply: emeasurable_fun_funeneg => //; exact: measurable_fun_prod2. + by apply: measurable_fun_comp => //; exact: measurable_fun_prod2. + by move=> x _; rewrite gee0_abs// -/((abse \o f) (x, y)) fune_abse lee_addr. Qed. Lemma fubini1 : \int[m1]_x F x = \int[m]_z f z. Proof. rewrite FE integralB// [in RHS]integralE//. rewrite fubini_tonelli1//; last exact: emeasurable_fun_funepos. by rewrite fubini_tonelli1//; exact: emeasurable_fun_funeneg. Qed. Lemma fubini2 : \int[m2]_x G x = \int[m]_z f z. Proof. rewrite GE integralB// [in RHS]integralE//. rewrite fubini_tonelli2//; last exact: emeasurable_fun_funepos. by rewrite fubini_tonelli2//; exact: emeasurable_fun_funeneg. Qed. Theorem Fubini : \int[m1]_x \int[m2]_y f (x, y) = \int[m2]_y \int[m1]_x f (x, y). Proof. by rewrite fubini1 -fubini2. Qed. End fubini.