formal/coq/analysis/lebesgue_measure.v

(* mathcomp analysis (c) 2017 Inria and AIST. License: CeCILL-C.              *)
From mathcomp Require Import all_ssreflect ssralg ssrnum ssrint interval.
From mathcomp Require Import finmap fingroup perm rat.
Require Import boolp reals ereal classical_sets signed topology numfun.
Require Import mathcomp_extra functions normedtype.
From HB Require Import structures.
Require Import sequences esum measure fsbigop cardinality set_interval.
Require Import realfun.

(******************************************************************************)
(*                            Lebesgue Measure                                *)
(*                                                                            *)
(* This file contains a formalization of the Lebesgue measure using the       *)
(* Caratheodory's theorem available in measure.v and further develops the     *)
(* theory of measurable functions.                                            *)
(*                                                                            *)
(* Main reference:                                                            *)
(* - Daniel Li, Intégration et applications, 2016                             *)
(* - Achim Klenke, Probability Theory 2nd edition, 2014                       *)
(*                                                                            *)
(*             hlength A == length of the hull of the set of real numbers A   *)
(*                 ocitv == set of open-closed intervals ]x, y] where         *)
(*                            x and y are real numbers                        *)
(*      lebesgue_measure == the Lebesgue measure                              *)
(*                                                                            *)
(*              ps_infty == inductive definition of the powerset              *)
(*                          {0, {-oo}, {+oo}, {-oo,+oo}}                      *)
(*         emeasurable G == sigma-algebra over \bar R built out of the        *)
(*                          measurables G of a sigma-algebra over R           *)
(*     elebesgue_measure == the Lebesgue measure extended to \bar R           *)
(*                                                                            *)
(* The modules RGenOInfty, RGenInftyO, RGenCInfty, RGenOpens provide proofs   *)
(* of equivalence between the sigma-algebra generated by list of intervals    *)
(* and the sigma-algebras generated by open rays, closed rays, and open       *)
(* intervals.                                                                 *)
(*                                                                            *)
(* The modules ErealGenOInfty and ErealGenCInfty provide proofs of            *)
(* equivalence between emeasurable and the sigma-algebras generated open      *)
(* rays and closed rays.                                                      *)
(*                                                                            *)
(******************************************************************************)

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 "R .-ocitv" (at level 1, format "R .-ocitv").
Reserved Notation "R .-ocitv.-measurable"
 (at level 2, format "R .-ocitv.-measurable").

Section hlength.
Local Open Scope ereal_scope.
Variable R : realType.
Implicit Types i j : interval R.

Definition hlength (A : set R) : \bar R := let i := Rhull A in i.2 - i.1.

Lemma hlength0 : hlength (set0 : set R) = 0.
Proof. by rewrite /hlength Rhull0 /= subee. Qed.

Lemma hlength_singleton (r : R) : hlength `[r, r] = 0.
Proof.
rewrite /hlength /= asboolT// sup_itvcc//= asboolT//.
by rewrite asboolT inf_itvcc//= ?subee// inE.
Qed.

Lemma hlength_setT : hlength setT = +oo%E :> \bar R.
Proof. by rewrite /hlength RhullT. Qed.

Lemma hlength_itv i : hlength [set` i] = if i.2 > i.1 then i.2 - i.1 else 0.
Proof.
case: ltP => [/lt_ereal_bnd/neitvP i12|]; first by rewrite /hlength set_itvK.
rewrite le_eqVlt => /orP[|/lt_ereal_bnd i12]; last first.
  rewrite (_ : [set` i] = set0) ?hlength0//.
  by apply/eqP/negPn; rewrite -/(neitv _) neitvE -leNgt (ltW i12).
case: i => -[ba a|[|]] [bb b|[|]] //=.
- rewrite /= => /eqP[->{b}]; move: ba bb => -[] []; try
    by rewrite set_itvE hlength0.
  by rewrite hlength_singleton.
- by move=> _; rewrite set_itvE hlength0.
- by move=> _; rewrite set_itvE hlength0.
Qed.

Lemma hlength_finite_fin_num i : neitv i -> hlength [set` i] < +oo ->
  ((i.1 : \bar R) \is a fin_num) /\ ((i.2 : \bar R) \is a fin_num).
Proof.
move: i => [[ba a|[]] [bb b|[]]] /neitvP //=; do ?by rewrite ?set_itvE ?eqxx.
by move=> _; rewrite hlength_itv /= ltey.
by move=> _; rewrite hlength_itv /= ltNye.
by move=> _; rewrite hlength_itv.
Qed.

Lemma finite_hlengthE i : neitv i -> hlength [set` i] < +oo ->
  hlength [set` i] = (fine i.2)%:E - (fine i.1)%:E.
Proof.
move=> i0 ioo; have [ri1 ri2] := hlength_finite_fin_num i0 ioo.
rewrite !fineK// hlength_itv; case: ifPn => //.
rewrite -leNgt le_eqVlt => /predU1P[->|]; first by rewrite subee.
by move/lt_ereal_bnd/ltW; rewrite leNgt; move: i0 => /neitvP => ->.
Qed.

Lemma hlength_infty_bnd b r :
  hlength [set` Interval -oo%O (BSide b r)] = +oo :> \bar R.
Proof. by rewrite hlength_itv /= ltNye. Qed.

Lemma hlength_bnd_infty b r :
  hlength [set` Interval (BSide b r) +oo%O] = +oo :> \bar R.
Proof. by rewrite hlength_itv /= ltey. Qed.

Lemma pinfty_hlength i : hlength [set` i] = +oo ->
  (exists s r, i = Interval -oo%O (BSide s r) \/ i = Interval (BSide s r) +oo%O)
  \/ i = `]-oo, +oo[.
Proof.
rewrite hlength_itv; case: i => -[ba a|[]] [bb b|[]] //= => [|_|_|].
- by case: ifPn.
- by left; exists ba, a; right.
- by left; exists bb, b; left.
- by right.
Qed.

Lemma hlength_ge0 i : 0 <= hlength [set` i].
Proof.
rewrite hlength_itv; case: ifPn => //; case: (i.1 : \bar _) => [r| |].
- by rewrite suber_ge0//; exact: ltW.
- by rewrite ltNge leey.
- by case: (i.2 : \bar _) => //= [r _]; rewrite leey.
Qed.
Local Hint Extern 0 (0%:E <= hlength _) => solve[apply: hlength_ge0] : core.

Lemma hlength_Rhull (A : set R) : hlength [set` Rhull A] = hlength A.
Proof. by rewrite /hlength Rhull_involutive. Qed.

Lemma le_hlength_itv i j : {subset i <= j} -> hlength [set` i] <= hlength [set` j].
Proof.
set I := [set` i]; set J := [set` j].
have [->|/set0P I0] := eqVneq I set0; first by rewrite hlength0 hlength_ge0.
have [J0|/set0P J0] := eqVneq J set0.
  by move/subset_itvP; rewrite -/J J0 subset0 -/I => ->.
move=> /subset_itvP ij; apply: lee_sub => /=.
  have [ui|ui] := asboolP (has_ubound I).
    have [uj /=|uj] := asboolP (has_ubound J); last by rewrite leey.
    by rewrite lee_fin le_sup // => r Ir; exists r; split => //; apply: ij.
  have [uj /=|//] := asboolP (has_ubound J).
  by move: ui; have := subset_has_ubound ij uj.
have [lj /=|lj] := asboolP (has_lbound J); last by rewrite leNye.
have [li /=|li] := asboolP (has_lbound I); last first.
  by move: li; have := subset_has_lbound ij lj.
rewrite lee_fin ler_oppl opprK le_sup// ?has_inf_supN//; last exact/nonemptyN.
move=> r [r' Ir' <-{r}]; exists (- r')%R.
by split => //; exists r' => //; apply: ij.
Qed.

Lemma le_hlength : {homo hlength : A B / (A `<=` B) >-> A <= B}.
Proof.
move=> a b /le_Rhull /le_hlength_itv.
by rewrite (hlength_Rhull a) (hlength_Rhull b).
Qed.

End hlength.
Arguments hlength {R}.
#[global] Hint Extern 0 (0%:E <= hlength _) => solve[apply: hlength_ge0] : core.

Section itv_semiRingOfSets.
Variable R : realType.
Implicit Types (I J K : set R).
Definition ocitv_type : Type := R.

Definition ocitv := [set `]x.1, x.2]%classic | x in [set: R * R]].

Lemma is_ocitv a b : ocitv `]a, b]%classic.
Proof. by exists (a, b); split => //=; rewrite in_itv/= andbT. Qed.
Hint Extern 0 (ocitv _) => solve [apply: is_ocitv] : core.

Lemma ocitv0 : ocitv set0.
Proof. by exists (1, 0); rewrite //= set_itv_ge ?bnd_simp//= ltr10. Qed.
Hint Resolve ocitv0 : core.

Lemma ocitvP X : ocitv X <-> X = set0 \/ exists2 x, x.1 < x.2 & X = `]x.1, x.2]%classic.
Proof.
split=> [[x _ <-]|[->//|[x xlt ->]]]//.
case: (boolP (x.1 < x.2)) => x12; first by right; exists x.
by left; rewrite set_itv_ge.
Qed.

Lemma ocitvD : semi_setD_closed ocitv.
Proof.
move=> _ _ [a _ <-] /ocitvP[|[b ltb]] ->.
  rewrite setD0; exists [set `]a.1, a.2]%classic].
  by split=> [//|? ->//||? ? -> ->//]; rewrite bigcup_set1.
rewrite setDE setCitv/= setIUr -!set_itvI.
rewrite /Order.meet/= /Order.meet/= /Order.join/=
         ?(andbF, orbF)/= ?(meetEtotal, joinEtotal).
rewrite -negb_or le_total/=; set c := minr _ _; set d := maxr _ _.
have inside : a.1 < c -> d < a.2 -> `]a.1, c] `&` `]d, a.2] = set0.
  rewrite -subset0 lt_minr lt_maxl => /andP[a12 ab1] /andP[_ ba2] x /= [].
  have b1a2 : b.1 <= a.2 by rewrite ltW// (lt_trans ltb).
  have a1b2 : a.1 <= b.2 by rewrite ltW// (lt_trans _ ltb).
  rewrite /c /d (min_idPr _)// (max_idPr _)// !in_itv /=.
  move=> /andP[a1x xb1] /andP[b2x xa2].
  by have := lt_le_trans b2x xb1; case: ltgtP ltb.
exists ((if a.1 < c then [set `]a.1, c]%classic] else set0) `|`
        (if d < a.2 then [set `]d, a.2]%classic] else set0)); split.
- by rewrite finite_setU; do! case: ifP.
- by move=> ? []; case: ifP => ? // ->//=.
- by rewrite bigcup_setU; congr (_ `|` _);
     case: ifPn => ?; rewrite ?bigcup_set1 ?bigcup_set0// set_itv_ge.
- move=> I J/=; case: ifP => //= ac; case: ifP => //= da [] // -> []// ->.
    by rewrite inside// => -[].
  by rewrite setIC inside// => -[].
Qed.

Lemma ocitvI : setI_closed ocitv.
Proof.
move=> _ _ [a _ <-] [b _ <-]; rewrite -set_itvI/=.
rewrite /Order.meet/= /Order.meet /Order.join/=
        ?(andbF, orbF)/= ?(meetEtotal, joinEtotal).
by rewrite -negb_or le_total/=.
Qed.

Definition ocitv_display : Type -> measure_display. Proof. exact. Qed.

HB.instance Definition _ :=
  @isSemiRingOfSets.Build (ocitv_display R)
    ocitv_type (Pointed.class R) ocitv ocitv0 ocitvI ocitvD.

Notation "R .-ocitv" := (ocitv_display R) : measure_display_scope.
Notation "R .-ocitv.-measurable" := (measurable : set (set (ocitv_type))) :
  classical_set_scope.

Lemma hlength_ge0' (I : set ocitv_type) : (0 <= hlength I)%E.
Proof. by rewrite -hlength0 le_hlength. Qed.

(* Unused *)
(* Lemma hlength_semi_additive2 : semi_additive2 hlength. *)
(* Proof. *)
(* move=> I J /ocitvP[|[a a12]] ->; first by rewrite set0U hlength0 add0e. *)
(* move=> /ocitvP[|[b b12]] ->; first by rewrite setU0 hlength0 adde0. *)
(* rewrite -subset0 => + ab0 => /ocitvP[|[x x12] abx]. *)
(*   by rewrite setU_eq0 => -[-> ->]; rewrite setU0 hlength0 adde0. *)
(* rewrite abx !hlength_itv//= ?lte_fin a12 b12 x12/= -!EFinB -EFinD. *)
(* wlog ab1 : a a12 b b12 ab0 abx / a.1 <= b.1 => [hwlog|]. *)
(*   have /orP[ab1|ba1] := le_total a.1 b.1; first by apply: hwlog. *)
(*   by rewrite [in RHS]addrC; apply: hwlog => //; rewrite (setIC, setUC). *)
(* have := ab0; rewrite subset0 -set_itv_meet/=. *)
(* rewrite /Order.join /Order.meet/= ?(andbF, orbF)/= ?(meetEtotal, joinEtotal). *)
(* rewrite -negb_or le_total/=; set c := minr _ _; set d := maxr _ _. *)
(* move=> /eqP/neitvP/=; rewrite bnd_simp/= /d/c (max_idPr _)// => /negP. *)
(* rewrite -leNgt le_minl orbC lt_geF//= => {c} {d} a2b1. *)
(* have ab i j : i \in `]a.1, a.2] -> j \in `]b.1, b.2] -> i <= j. *)
(*   by move=> ia jb; rewrite (le_le_trans _ _ a2b1) ?(itvP ia) ?(itvP jb). *)
(* have /(congr1 sup) := abx; rewrite sup_setU// ?sup_itv_bounded// => bx. *)
(* have /(congr1 inf) := abx; rewrite inf_setU// ?inf_itv_bounded// => ax. *)
(* rewrite -{}ax -{x}bx in abx x12 *. *)
(* case: ltgtP a2b1 => // a2b1 _; last first. *)
(*   by rewrite a2b1 [in RHS]addrC subrKA. *)
(* exfalso; pose c := (a.2 + b.1) / 2%:R. *)
(* have /predeqP/(_ c)[_ /(_ _)/Box[]] := abx. *)
(*   apply: subset_itv_oo_oc; have := mid_in_itvoo a2b1. *)
(*   by apply/subitvP; rewrite subitvE ?bnd_simp/= ?ltW. *)
(* apply/not_orP; rewrite /= !in_itv/=. *)
(* by rewrite lt_geF ?midf_lt//= andbF le_gtF ?midf_le//= ltW. *)
(* Qed. *)

Lemma hlength_semi_additive : semi_additive (hlength : set ocitv_type -> _).
Proof.
move=> /= I n /(_ _)/cid2-/all_sig[b]/all_and2[_]/(_ _)/esym-/funext {I}->.
move=> Itriv [[/= a1 a2] _] /esym /[dup] + ->.
rewrite hlength_itv ?lte_fin/= -EFinB.
case: ifPn => a12; last first.
  pose I i :=  `](b i).1, (b i).2]%classic.
  rewrite set_itv_ge//= -(bigcup_mkord _ I) /I => /bigcup0P I0.
  by under eq_bigr => i _ do rewrite I0//= hlength0; rewrite big1.
set A := `]a1, a2]%classic.
rewrite -bigcup_pred; set P := xpredT; rewrite (eq_bigl P)//.
move: P => P; have [p] := ubnP #|P|; elim: p => // p IHp in P a2 a12 A *.
rewrite ltnS => cP /esym AE.
have : A a2 by rewrite /A /= in_itv/= lexx andbT.
rewrite AE/= => -[i /= Pi] a2bi.
case: (boolP ((b i).1 < (b i).2)) => bi; last by rewrite itv_ge in a2bi.
have {}a2bi : a2 = (b i).2.
  apply/eqP; rewrite eq_le (itvP a2bi)/=.
  suff: A (b i).2 by move=> /itvP->.
  by rewrite AE; exists i=> //=; rewrite in_itv/= lexx andbT.
rewrite {a2}a2bi in a12 A AE *.
rewrite (bigD1 i)//= hlength_itv ?lte_fin/= bi !EFinD -addeA.
congr (_ + _)%E; apply/eqP; rewrite addeC -sube_eq// 1?adde_defC//.
rewrite ?EFinN oppeK addeC; apply/eqP.
case: (eqVneq a1 (b i).1) => a1bi.
  rewrite {a1}a1bi in a12 A AE {IHp} *; rewrite subee ?big1// => j.
  move=> /andP[Pj Nji]; rewrite hlength_itv ?lte_fin/=; case: ifPn => bj//.
  exfalso; have /trivIsetP/(_ j i I I Nji) := Itriv.
  pose m := ((b j).1 + (b j).2) / 2%:R.
  have mbj : `](b j).1, (b j).2]%classic m.
     by rewrite /= !in_itv/= ?(midf_lt, midf_le)//= ltW.
  rewrite -subset0 => /(_ m); apply; split=> //.
  by suff: A m by []; rewrite AE; exists j => //.
have a1b2 j : P j -> (b j).1 < (b j).2 -> a1 <= (b j).2.
  move=> Pj bj; suff /itvP-> : A (b j).2 by [].
  by rewrite AE; exists j => //=; rewrite ?in_itv/= bj//=.
have a1b j : P j -> (b j).1 < (b j).2 -> a1 <= (b j).1.
  move=> Pj bj; case: ltP=> // bj1a.
  suff : A a1 by rewrite /A/= in_itv/= ltxx.
  by rewrite AE; exists j; rewrite //= in_itv/= bj1a//= a1b2.
have bbi2 j : P j -> (b j).1 < (b j).2 -> (b j).2 <= (b i).2.
  move=> Pj bj; suff /itvP-> : A (b j).2 by [].
  by rewrite AE; exists j => //=; rewrite ?in_itv/= bj//=.
apply/IHp.
- by rewrite lt_neqAle a1bi/= a1b.
- rewrite (leq_trans _ cP)// -(cardID (pred1 i) P).
  rewrite [X in (_ < X + _)%N](@eq_card _ _ (pred1 i)); last first.
    by move=> j; rewrite !inE andbC; case: eqVneq => // ->.
  rewrite ?card1 ?ltnS// subset_leq_card//.
  by apply/fintype.subsetP => j; rewrite -topredE/= !inE andbC.
apply/seteqP; split=> /= [x [j/= /andP[Pj Nji]]|x/= xabi].
  case: (boolP ((b j).1 < (b j).2)) => bj; last by rewrite itv_ge.
  apply: subitvP; rewrite subitvE ?bnd_simp a1b//= leNgt.
  have /trivIsetP/(_ j i I I Nji) := Itriv.
  rewrite -subset0 => /(_ (b j).2); apply: contra_notN => /= bi1j2.
  by rewrite !in_itv/= bj !lexx bi1j2 bbi2.
have: A x.
  rewrite /A/= in_itv/= (itvP xabi)/= ltW//.
  by rewrite (le_lt_trans _ bi) ?(itvP xabi).
rewrite AE => -[j /= Pj xbj].
exists j => //=.
apply/andP; split=> //; apply: contraTneq xbj => ->.
by rewrite in_itv/= le_gtF// (itvP xabi).
Qed.

HB.instance Definition _ := isAdditiveMeasure.Build _ R _
  (hlength : set ocitv_type -> _) (@hlength_ge0') hlength_semi_additive.

Hint Extern 0 ((_ .-ocitv).-measurable _) => solve [apply: is_ocitv] : core.

Lemma hlength_sigma_sub_additive :
  sigma_sub_additive (hlength : set ocitv_type -> _).
Proof.
move=> I A /(_ _)/cid2-/all_sig[b]/all_and2[_]/(_ _)/esym AE.
move=> [a _ <-]; rewrite hlength_itv ?lte_fin/= -EFinB => lebig.
case: ifPn => a12; last by rewrite nneseries_esum// esum_ge0.
apply: lee_adde => e.
rewrite [e%:num]splitr [in leRHS]EFinD addeA -lee_subl_addr//.
apply: le_trans (epsilon_trick _ _ _) => //=.
have eVn_gt0 n : 0 < e%:num / 2 / (2 ^ n.+1)%:R.
  by rewrite divr_gt0// ltr0n// expn_gt0.
have eVn_ge0 n := ltW (eVn_gt0 n).
pose Aoo i : set ocitv_type :=
  `](b i).1, (b i).2 + e%:num / 2 / (2 ^ i.+1)%:R[%classic.
pose Aoc i : set ocitv_type :=
  `](b i).1, (b i).2 + e%:num / 2 / (2 ^ i.+1)%:R]%classic.
have: `[a.1 + e%:num / 2, a.2] `<=` \bigcup_i Aoo i.
  apply: (@subset_trans _ `]a.1, a.2]).
    move=> x; rewrite /= !in_itv /= => /andP[+ -> //].
    by move=> /lt_le_trans-> //; rewrite ltr_addl.
  apply: (subset_trans lebig); apply: subset_bigcup => i _; rewrite AE /Aoo/=.
  move=> x /=; rewrite !in_itv /= => /andP[-> /le_lt_trans->]//=.
  by rewrite ltr_addl.
have := @segment_compact _ (a.1 + e%:num / 2) a.2; rewrite compact_cover.
move=> /[apply]-[i _|X _ Xc]; first exact: interval_open.
have: `](a.1 + e%:num / 2), a.2] `<=` \bigcup_(i in [set` X]) Aoc i.
  move=> x /subset_itv_oc_cc /Xc [i /= Xi] Aooix.
  by exists i => //; apply: subset_itv_oo_oc Aooix.
have /[apply] := @content_sub_fsum _ _ _
  [the additive_measure _ _ of hlength : set ocitv_type -> _] _ [set` X].
move=> /(_ _ _ _)/Box[]//=; apply: le_le_trans.
  rewrite hlength_itv ?lte_fin -?EFinD/= -addrA -opprD.
  by case: ltP => //; rewrite lee_fin subr_le0.
rewrite nneseries_esum//; last by move=> *; rewrite adde_ge0//= ?lee_fin.
rewrite esum_ge//; exists X => //; rewrite fsbig_finite// ?set_fsetK//=.
rewrite lee_sum // => i _; rewrite ?AE// !hlength_itv/= ?lte_fin -?EFinD/=.
do !case: ifPn => //= ?; do ?by rewrite ?adde_ge0 ?lee_fin// ?subr_ge0// ?ltW.
  by rewrite addrAC.
by rewrite addrAC lee_fin ler_add// subr_le0 leNgt.
Qed.

Lemma hlength_sigma_finite : sigma_finite [set: ocitv_type] hlength.
Proof.
exists (fun k : nat => `] (- k%:R)%R, k%:R]%classic).
  apply/esym; rewrite -subTset => /= x _ /=.
  exists `|(floor `|x|%R + 1)%R|%N; rewrite //= in_itv/=.
  rewrite !natr_absz intr_norm intrD -RfloorE.
  suff: `|x| < `|Rfloor `|x| + 1| by rewrite ltr_norml => /andP[-> /ltW->].
  rewrite [ltRHS]ger0_norm//.
    by rewrite (le_lt_trans _ (lt_succ_Rfloor _))// ?ler_norm.
  by rewrite addr_ge0// -Rfloor0 le_Rfloor.
by move=> k; split => //; rewrite hlength_itv/= -EFinB; case: ifP; rewrite ltey.
Qed.

Let gitvs := [the semiRingOfSetsType _ of salgebraType ocitv].

Definition lebesgue_measure := Hahn_ext
  [the additive_measure _ _ of hlength : set ocitv_type -> _].

Let lebesgue_measure0 : lebesgue_measure set0 = 0%E.
Proof. by []. Qed.

Let lebesgue_measure_ge0 : forall x, (0 <= lebesgue_measure x)%E.
Proof. exact: measure.Hahn_ext_ge0. Qed.

Let lebesgue_measure_semi_sigma_additive : semi_sigma_additive lebesgue_measure.
Proof. exact/measure.Hahn_ext_sigma_additive/hlength_sigma_sub_additive. Qed.

HB.instance Definition _ := isMeasure.Build _ _ _ lebesgue_measure
  lebesgue_measure0 lebesgue_measure_ge0 lebesgue_measure_semi_sigma_additive.

End itv_semiRingOfSets.
Arguments lebesgue_measure {R}.

Notation "R .-ocitv" := (ocitv_display R) : measure_display_scope.
Notation "R .-ocitv.-measurable" := (measurable : set (set (ocitv_type R))) :
  classical_set_scope.

Section lebesgue_measure.
Variable R : realType.
Let gitvs := [the measurableType _ of salgebraType (@ocitv R)].

Lemma lebesgue_measure_unique (mu : {measure set gitvs -> \bar R}) :
  (forall X, ocitv X -> hlength X = mu X) ->
  forall X, measurable X -> lebesgue_measure X = mu X.
Proof.
move=> muE X mX; apply: Hahn_ext_unique => //=.
- exact: hlength_sigma_sub_additive.
- exact: hlength_sigma_finite.
Qed.

End lebesgue_measure.

Section ps_infty.
Context {T : Type}.
Local Open Scope ereal_scope.

Inductive ps_infty : set \bar T -> Prop :=
| ps_infty0 : ps_infty set0
| ps_ninfty : ps_infty [set -oo]
| ps_pinfty : ps_infty [set +oo]
| ps_inftys : ps_infty [set -oo; +oo].

Lemma ps_inftyP (A : set \bar T) : ps_infty A <-> A `<=` [set -oo; +oo].
Proof.
split => [[]//|Aoo].
by have [] := subset_set2 Aoo; move=> ->; constructor.
Qed.

Lemma setCU_Efin (A : set T) (B : set \bar T) : ps_infty B ->
  ~` (EFin @` A) `&` ~` B = (EFin @` ~` A) `|` ([set -oo%E; +oo%E] `&` ~` B).
Proof.
move=> ps_inftyB.
have -> : ~` (EFin @` A) = EFin @` (~` A) `|` [set -oo; +oo]%E.
  by rewrite EFin_setC setDKU // => x [|] -> -[].
rewrite setIUl; congr (_ `|` _); rewrite predeqE => -[x| |]; split; try by case.
by move=> [] x' Ax' [] <-{x}; split; [exists x'|case: ps_inftyB => // -[]].
Qed.

End ps_infty.

Section salgebra_ereal.
Variables (R : realType) (G : set (set R)).
Let measurableR : set (set R) := G.-sigma.-measurable.

Definition emeasurable : set (set \bar R) :=
  [set EFin @` A `|` B | A in measurableR & B in ps_infty].

Lemma emeasurable0 : emeasurable set0.
Proof.
exists set0; first exact: measurable0.
by exists set0; rewrite ?setU0// ?image_set0//; constructor.
Qed.

Lemma emeasurableC (X : set \bar R) : emeasurable X -> emeasurable (~` X).
Proof.
move => -[A mA] [B PooB <-]; rewrite setCU setCU_Efin //.
exists (~` A); [exact: measurableC | exists ([set -oo%E; +oo%E] `&` ~` B) => //].
case: PooB.
- by rewrite setC0 setIT; constructor.
- rewrite setIUl setICr set0U -setDE.
  have [_ ->] := @setDidPl (\bar R) [set +oo%E] [set -oo%E]; first by constructor.
  by rewrite predeqE => x; split => // -[->].
- rewrite setIUl setICr setU0 -setDE.
  have [_ ->] := @setDidPl (\bar R) [set -oo%E] [set +oo%E]; first by constructor.
  by rewrite predeqE => x; split => // -[->].
- by rewrite setICr; constructor.
Qed.

Lemma bigcupT_emeasurable (F : (set \bar R)^nat) :
  (forall i, emeasurable (F i)) -> emeasurable (\bigcup_i (F i)).
Proof.
move=> mF; pose P := fun i j => measurableR j.1 /\ ps_infty j.2 /\
                            F i = [set x%:E | x in j.1] `|` j.2.
have [f fi] : {f : nat -> (set R) * (set \bar R) & forall i, P i (f i) }.
  by apply: choice => i; have [x mx [y PSoo'y] xy] := mF i; exists (x, y).
exists (\bigcup_i (f i).1).
  by apply: bigcupT_measurable => i; exact: (fi i).1.
exists (\bigcup_i (f i).2).
  apply/ps_inftyP => x [n _] fn2x.
  have /ps_inftyP : ps_infty(f n).2 by have [_ []] := fi n.
  exact.
rewrite [RHS](@eq_bigcupr _ _ _ _
    (fun i => [set x%:E | x in (f i).1] `|` (f i).2)); last first.
  by move=> i; have [_ []] := fi i.
rewrite bigcupU; congr (_ `|` _).
rewrite predeqE => i /=; split=> [[r [n _ fn1r <-{i}]]|[n _ [r fn1r <-{i}]]];
 by [exists n => //; exists r | exists r => //; exists n].
Qed.

Definition ereal_isMeasurable :
  isMeasurable default_measure_display (\bar R) :=
  isMeasurable.Build _ _ (Pointed.class _)
    emeasurable0 emeasurableC bigcupT_emeasurable.

End salgebra_ereal.

Section puncture_ereal_itv.
Variable R : realDomainType.
Implicit Types (y : R) (b : bool).
Local Open Scope ereal_scope.

Lemma punct_eitv_bnd_pinfty b y : [set` Interval (BSide b y%:E) +oo%O] =
  EFin @` [set` Interval (BSide b y) +oo%O] `|` [set +oo].
Proof.
rewrite predeqE => x; split; rewrite /= in_itv andbT.
- move: x => [x| |] yxb; [|by right|by case: b yxb].
  by left; exists x => //; rewrite in_itv /= andbT; case: b yxb.
- move=> [[r]|->].
  + by rewrite in_itv /= andbT => yxb <-; case: b yxb.
  + by case: b => /=; rewrite ?(ltey, leey).
Qed.

Lemma punct_eitv_ninfty_bnd b y : [set` Interval -oo%O (BSide b y%:E)] =
  [set -oo%E] `|` EFin @` [set x | x \in Interval -oo%O (BSide b y)].
Proof.
rewrite predeqE => x; split; rewrite /= in_itv.
- move: x => [x| |] yxb; [|by case: b yxb|by left].
  by right; exists x => //; rewrite in_itv /= andbT; case: b yxb.
- move=> [->|[r]].
  + by case: b => /=; rewrite ?(ltNye, leNye).
  + by rewrite in_itv /= => yxb <-; case: b yxb.
Qed.

Lemma punct_eitv_setTR : range (@EFin R) `|` [set +oo] = [set~ -oo].
Proof.
rewrite eqEsubset; split => [a [[a' _ <-]|->]|] //.
by move=> [x| |] //= _; [left; exists x|right].
Qed.

Lemma punct_eitv_setTL : range (@EFin R) `|` [set -oo] = [set~ +oo].
Proof.
rewrite eqEsubset; split => [a [[a' _ <-]|->]|] //.
by move=> [x| |] //= _; [left; exists x|right].
Qed.

End puncture_ereal_itv.

Lemma set1_bigcap_oc (R : realType) (r : R) :
   [set r] = \bigcap_i `]r - i.+1%:R^-1, r]%classic.
Proof.
apply/seteqP; split=> [x ->|].
  by move=> i _/=; rewrite in_itv/= lexx ltr_subl_addr ltr_addl invr_gt0 ltr0n.
move=> x rx; apply/esym/eqP; rewrite eq_le (itvP (rx 0%N _))// andbT.
apply/ler_addgt0Pl => e e_gt0; rewrite -ler_subl_addl ltW//.
have := rx `|floor e^-1%R|%N I; rewrite /= in_itv => /andP[/le_lt_trans->]//.
rewrite ler_add2l ler_opp2 -lef_pinv ?invrK//; last by rewrite qualifE.
rewrite -addn1 natrD natr_absz ger0_norm ?floor_ge0 ?invr_ge0 1?ltW//.
by rewrite -RfloorE lt_succ_Rfloor.
Qed.

Lemma itv_bnd_open_bigcup (R : realType) b (r s : R) :
  [set` Interval (BSide b r) (BLeft s)] =
  \bigcup_n [set` Interval (BSide b r) (BRight (s - n.+1%:R^-1))].
Proof.
apply/seteqP; split => [x/=|]; last first.
  move=> x [n _ /=] /[!in_itv] /andP[-> /le_lt_trans]; apply.
  by rewrite ltr_subl_addr ltr_addl invr_gt0 ltr0n.
rewrite in_itv/= => /andP[sx xs]; exists `|ceil ((s - x)^-1)|%N => //=.
rewrite in_itv/= sx/= ler_subr_addl addrC -ler_subr_addl.
rewrite -[in X in _ <= X](invrK (s - x)) ler_pinv.
- rewrite -addn1 natrD natr_absz ger0_norm; last first.
    by rewrite ceil_ge0// invr_ge0 subr_ge0 ltW.
  by rewrite (@le_trans _ _ (ceil (s - x)^-1)%:~R)// ?ler_addl// ceil_ge.
- by rewrite inE unitfE ltr0n andbT pnatr_eq0.
- by rewrite inE invr_gt0 subr_gt0 xs andbT unitfE invr_eq0 subr_eq0 gt_eqF.
Qed.

Lemma itv_open_bnd_bigcup (R : realType) b (r s : R) :
  [set` Interval (BRight s) (BSide b r)] =
  \bigcup_n [set` Interval (BLeft (s + n.+1%:R^-1)) (BSide b r)].
Proof.
have /(congr1 (fun x => -%R @` x)) := itv_bnd_open_bigcup (~~ b) (- r) (- s).
rewrite opp_itv_bnd_bnd/= !opprK negbK => ->; rewrite image_bigcup.
apply eq_bigcupr => k _; apply/seteqP; split=> [_/= [y ysr] <-|x/= xsr].
  by rewrite oppr_itv/= opprD.
by exists (- x); rewrite ?oppr_itv//= opprK// negbK opprB opprK addrC.
Qed.

Lemma itv_bnd_infty_bigcup (R : realType) b (x : R) :
  [set` Interval (BSide b x) +oo%O] =
  \bigcup_i [set` Interval (BSide b x) (BRight (x + i%:R))].
Proof.
apply/seteqP; split=> y; rewrite /= !in_itv/= andbT; last first.
  by move=> [k _ /=]; move: b => [|] /=; rewrite in_itv/= => /andP[//] /ltW.
move=> xy; exists `|ceil (y - x)|%N => //=; rewrite in_itv/= xy/= -ler_subl_addl.
rewrite !natr_absz/= ger0_norm ?ceil_ge0// ?subr_ge0//; last first.
  by case: b xy => //= /ltW.
by rewrite -RceilE Rceil_ge.
Qed.

Lemma itv_infty_bnd_bigcup (R : realType) b (x : R) :
  [set` Interval -oo%O (BSide b x)] =
  \bigcup_i [set` Interval (BLeft (x - i%:R)) (BSide b x)].
Proof.
have /(congr1 (fun x => -%R @` x)) := itv_bnd_infty_bigcup (~~ b) (- x).
rewrite opp_itv_bnd_infty negbK opprK => ->; rewrite image_bigcup.
apply eq_bigcupr => k _; apply/seteqP; split=> [_ /= -[r rbxk <-]|y/= yxkb].
   by rewrite oppr_itv/= opprB addrC.
by exists (- y); [rewrite oppr_itv/= negbK opprD opprK|rewrite opprK].
Qed.

Section salgebra_R_ssets.
Variable R : realType.

Definition measurableTypeR := salgebraType (R.-ocitv.-measurable).
Definition measurableR : set (set R) :=
  (R.-ocitv.-measurable).-sigma.-measurable.

HB.instance Definition R_isMeasurable :
  isMeasurable default_measure_display R :=
  @isMeasurable.Build _ measurableTypeR (Pointed.class R) measurableR
    measurable0 (@measurableC _ _) (@bigcupT_measurable _ _).
(*HB.instance (Real.sort R) R_isMeasurable.*)

Lemma measurable_set1 (r : R) : measurable [set r].
Proof.
rewrite set1_bigcap_oc; apply: bigcap_measurable => k // _.
by apply: sub_sigma_algebra; exact/is_ocitv.
Qed.
#[local] Hint Resolve measurable_set1 : core.

Lemma measurable_itv (i : interval R) : measurable [set` i].
Proof.
have moc (a b : R) : measurable `]a, b]%classic.
  by apply: sub_sigma_algebra; apply: is_ocitv.
have mopoo (x : R) : measurable `]x, +oo[%classic.
  by rewrite itv_bnd_infty_bigcup; exact: bigcup_measurable.
have mnooc (x : R) : measurable `]-oo, x]%classic.
  by rewrite -setCitvr; exact/measurableC.
have ooE (a b : R) : `]a, b[%classic = `]a, b]%classic `\ b.
  case: (boolP (a < b)) => ab; last by rewrite !set_itv_ge ?set0D.
  by rewrite -setUitv1// setUDK// => x [->]; rewrite /= in_itv/= ltxx andbF.
have moo (a b : R) : measurable `]a, b[%classic.
  by rewrite ooE; exact: measurableD.
have mcc (a b : R) : measurable `[a, b]%classic.
  case: (boolP (a <= b)) => ab; last by rewrite set_itv_ge.
  by rewrite -setU1itv//; apply/measurableU.
have mco (a b : R) : measurable `[a, b[%classic.
  case: (boolP (a < b)) => ab; last by rewrite set_itv_ge.
  by rewrite -setU1itv//; apply/measurableU.
have oooE (b : R) : `]-oo, b[%classic = `]-oo, b]%classic `\ b.
  by rewrite -setUitv1// setUDK// => x [->]; rewrite /= in_itv/= ltxx.
case: i => [[[] a|[]] [[] b|[]]] => //; do ?by rewrite set_itv_ge.
- by rewrite -setU1itv//; exact/measurableU.
- by rewrite oooE; exact/measurableD.
- by rewrite set_itv_infty_infty.
Qed.

HB.instance Definition _ :=
  (ereal_isMeasurable (R.-ocitv.-measurable)).
(* NB: Until we dropped support for Coq 8.12, we were using
HB.instance (\bar (Real.sort R))
  (ereal_isMeasurable (@measurable (@itvs_semiRingOfSets R))).
This was producing a warning but the alternative was failing with Coq 8.12 with
  the following message (according to the CI):
  # [redundant-canonical-projection,typechecker]
  # forall (T : measurableType) (f : T -> R), measurable_fun setT f
  #      : Prop
  # File "./theories/lebesgue_measure.v", line 4508, characters 0-88:
  # Error: Anomaly "Uncaught exception Failure("sep_last")."
  # Please report at http://coq.inria.fr/bugs/.
*)

Lemma measurable_EFin (A : set R) : measurableR A -> measurable (EFin @` A).
Proof.
by move=> mA; exists A => //; exists set0; [constructor|rewrite setU0].
Qed.

Lemma emeasurable_set1 (x : \bar R) : measurable [set x].
Proof.
case: x => [r| |].
- by rewrite -image_set1; apply: measurable_EFin; apply: measurable_set1.
- exists set0 => //; [exists [set +oo%E]; [by constructor|]].
  by rewrite image_set0 set0U.
- exists set0 => //; [exists [set -oo%E]; [by constructor|]].
  by rewrite image_set0 set0U.
Qed.
#[local] Hint Resolve emeasurable_set1 : core.

Lemma itv_cpinfty_pinfty : `[+oo%E, +oo[%classic = [set +oo%E] :> set (\bar R).
Proof.
by rewrite set_itvE predeqE => t; split => /= [|<-//]; rewrite leye_eq => /eqP.
Qed.

Lemma itv_opinfty_pinfty : `]+oo%E, +oo[%classic = set0 :> set (\bar R).
Proof.
by rewrite set_itvE predeqE => t; split => //=; apply/negP; rewrite -leNgt leey.
Qed.

Lemma itv_cninfty_pinfty : `[-oo%E, +oo[%classic = setT :> set (\bar R).
Proof. by rewrite set_itvE predeqE => t; split => //= _; rewrite leNye. Qed.

Lemma itv_oninfty_pinfty :
  `]-oo%E, +oo[%classic = ~` [set -oo]%E :> set (\bar R).
Proof.
rewrite set_itvE predeqE => x; split => /=.
- by move: x => [x| |]; rewrite ?ltxx.
- by move: x => [x h|//|/(_ erefl)]; rewrite ?ltNye.
Qed.

Lemma emeasurable_itv_bnd_pinfty b (y : \bar R) :
  measurable [set` Interval (BSide b y) +oo%O].
Proof.
move: y => [y| |].
- exists [set` Interval (BSide b y) +oo%O]; first exact: measurable_itv.
  by exists [set +oo%E]; [constructor|rewrite -punct_eitv_bnd_pinfty].
- by case: b; rewrite ?itv_opinfty_pinfty ?itv_cpinfty_pinfty.
- case: b; first by rewrite itv_cninfty_pinfty.
  by rewrite itv_oninfty_pinfty; exact/measurableC.
Qed.

Lemma emeasurable_itv_ninfty_bnd b (y : \bar R) :
  measurable [set` Interval -oo%O (BSide b y)].
Proof.
by rewrite -setCitvr; exact/measurableC/emeasurable_itv_bnd_pinfty.
Qed.

Definition elebesgue_measure : set \bar R -> \bar R :=
  fun S => lebesgue_measure (fine @` (S `\` [set -oo; +oo]%E)).

Lemma elebesgue_measure0 : elebesgue_measure set0 = 0%E.
Proof. by rewrite /elebesgue_measure set0D image_set0 measure0. Qed.

Lemma measurable_fine (X : set \bar R) : measurable X ->
  measurable [set fine x | x in X `\` [set -oo; +oo]%E].
Proof.
case => Y mY [X' [ | <-{X} | <-{X} | <-{X} ]].
- rewrite setU0 => <-{X}.
  rewrite [X in measurable X](_ : _ = Y) // predeqE => r; split.
    by move=> [x [[x' Yx' <-{x}/= _ <-//]]].
  by move=> Yr; exists r%:E; split => [|[]//]; exists r.
- rewrite [X in measurable X](_ : _ = Y) // predeqE => r; split.
    move=> [x [[[x' Yx' <- _ <-//]|]]].
    by move=> <-; rewrite not_orP => -[]/(_ erefl).
  by move=> Yr; exists r%:E => //; split => [|[]//]; left; exists r.
- rewrite [X in measurable X](_ : _ = Y) // predeqE => r; split.
    move=> [x [[[x' Yx' <-{x} _ <-//]|]]].
    by move=> ->; rewrite not_orP => -[_]/(_ erefl).
  by move=> Yr; exists r%:E => //; split => [|[]//]; left; exists r.
- rewrite [X in measurable X](_ : _ = Y) // predeqE => r; split.
    by rewrite setDUl setDv setU0 => -[_ [[x' Yx' <-]] _ <-].
  by move=> Yr; exists r%:E => //; split => [|[]//]; left; exists r.
Qed.

Lemma elebesgue_measure_ge0 X : (0 <= elebesgue_measure X)%E.
Proof. exact/measure_ge0. Qed.

Lemma semi_sigma_additive_elebesgue_measure :
  semi_sigma_additive elebesgue_measure.
Proof.
move=> /= F mF tF mUF; rewrite /elebesgue_measure.
rewrite [X in lebesgue_measure X](_ : _ =
    \bigcup_n (fine @` (F n `\` [set -oo; +oo]%E))); last first.
  rewrite predeqE => r; split.
    by move=> [x [[n _ Fnx xoo <-]]]; exists n => //; exists x.
  by move=> [n _ [x [Fnx xoo <-{r}]]]; exists x => //; split => //; exists n.
apply: (@measure_semi_sigma_additive _ _ _ [the measure _ _ of (@lebesgue_measure R)]
  (fun n => fine @` (F n `\` [set -oo; +oo]%E))).
- move=> n; have := mF n.
  move=> [X mX [X' mX']] XX'Fn.
  apply: measurable_fine.
  rewrite -XX'Fn.
  apply: measurableU; first exact: measurable_EFin.
  by case: mX' => //; exact: measurableU.
- move=> i j _ _ [x [[a [Fia aoo ax] [b [Fjb boo] bx]]]].
  move: tF => /(_ i j Logic.I Logic.I); apply.
  suff ab : a = b by exists a; split => //; rewrite ab.
  move: a b {Fia Fjb} aoo boo ax bx.
  move=> [a| |] [b| |] /=.
  + by move=> _ _ -> ->.
  + by move=> _; rewrite not_orP => -[_]/(_ erefl).
  + by move=> _; rewrite not_orP => -[]/(_ erefl).
  + by rewrite not_orP => -[_]/(_ erefl).
  + by rewrite not_orP => -[_]/(_ erefl).
  + by rewrite not_orP => -[_]/(_ erefl).
  + by rewrite not_orP => -[]/(_ erefl).
  + by rewrite not_orP => -[]/(_ erefl).
  + by rewrite not_orP => -[]/(_ erefl).
- move: mUF.
  rewrite {1}/measurable /emeasurable /= => -[X mX [Y []]] {Y}.
  - rewrite setU0 => h.
    rewrite [X in measurable X](_ : _ = X) // predeqE => r; split => [|Xr].
      move=> -[n _ [x [Fnx xoo <-{r}]]].
      have : (\bigcup_n F n) x by exists n.
      by rewrite -h => -[x' Xx' <-].
    have [n _ Fnr] : (\bigcup_n F n) r%:E by rewrite -h; exists r.
    by exists n => //; exists r%:E => //; split => //; case.
  - move=> h.
    rewrite [X in measurable X](_ : _ = X) // predeqE => r; split => [|Xr].
      move=> -[n _ [x [Fnx xoo <-]]].
      have : (\bigcup_n F n) x by exists n.
      by rewrite -h => -[[x' Xx' <-//]|xoo']; move/not_orP : xoo => -[].
    have [n _ Fnr] : (\bigcup_n F n) r%:E by rewrite -h; left; exists r.
    by exists n => //; exists r%:E => //; split => //; case.
  - (* NB: almost the same as the previous one, factorize?*)
    move=> h.
    rewrite [X in measurable X](_ : _ = X) // predeqE => r; split => [|Xr].
      move=> -[n _ [x [Fnx xoo <-]]].
      have : (\bigcup_n F n) x by exists n.
      by rewrite -h => -[[x' Xx' <-//]|xoo']; move/not_orP : xoo => -[].
    have [n _ Fnr] : (\bigcup_n F n) r%:E by rewrite -h; left; exists r.
    by exists n => //; exists r%:E => //; split => //; case.
  - move=> h.
    rewrite [X in measurable X](_ : _ = X) // predeqE => r; split => [|Xr].
      move=> -[n _ [x [Fnx xoo <-]]].
      have : (\bigcup_n F n) x by exists n.
      by rewrite -h => -[[x' Xx' <-//]|].
    have [n _ Fnr] : (\bigcup_n F n) r%:E by rewrite -h; left; exists r.
    by exists n => //; exists r%:E => //; split => //; case.
Qed.

HB.instance Definition _ := isMeasure.Build _ _ _ elebesgue_measure
  elebesgue_measure0 elebesgue_measure_ge0
  semi_sigma_additive_elebesgue_measure.

End salgebra_R_ssets.
#[global]
Hint Extern 0 (measurable [set _]) => solve [apply: measurable_set1|
                                            apply: emeasurable_set1] : core.

Section lebesgue_measure_itv.
Variable R : realType.

Let lebesgue_measure_itvoc (a b : R) :
  (lebesgue_measure (`]a, b] : set R) = hlength `]a, b])%classic.
Proof.
rewrite /lebesgue_measure/= /Hahn_ext measurable_mu_extE//; last first.
  by exists (a, b).
exact: hlength_sigma_sub_additive.
Qed.

Let lebesgue_measure_itvoo_subr1 (a : R) :
  lebesgue_measure (`]a - 1, a[%classic : set R) = 1%E.
Proof.
rewrite itv_bnd_open_bigcup//; transitivity (lim (lebesgue_measure \o
    (fun k => `]a - 1, a - k.+1%:R^-1]%classic : set R))).
  apply/esym/cvg_lim => //; apply: cvg_mu_inc.
  - by move=> ?; exact: measurable_itv.
  - by apply: bigcup_measurable => k _; exact: measurable_itv.
  - move=> n m nm; apply/subsetPset => x /=; rewrite !in_itv/= => /andP[->/=].
    by move/le_trans; apply; rewrite ler_sub// ler_pinv ?ler_nat//;
      rewrite inE ltr0n andbT unitfE.
rewrite (_ : _ \o _ = (fun n => (1 - n.+1%:R^-1)%:E)); last first.
  apply/funext => n /=; rewrite lebesgue_measure_itvoc.
  have [->|n0] := eqVneq n 0%N; first by rewrite invr1 subrr set_itvoc0.
  rewrite hlength_itv/= lte_fin ifT; last first.
    by rewrite ler_lt_sub// invr_lt1 ?unitfE// ltr1n ltnS lt0n.
  by rewrite !(EFinB,EFinN) oppeB// addeAC addeA subee// add0e.
apply/cvg_lim => //=; apply/ereal_cvg_real; split => /=; first exact: nearW.
apply/(@cvg_distP _ [pseudoMetricNormedZmodType R of R^o]) => _/posnumP[e].
rewrite !near_simpl; near=> n; rewrite opprB addrCA subrr addr0 ger0_norm//.
by near: n; exact: near_infty_natSinv_lt.
Unshelve. all: by end_near. Qed.

Lemma lebesgue_measure_set1 (a : R) : lebesgue_measure [set a] = 0%E.
Proof.
suff : (lebesgue_measure (`]a - 1, a]%classic%R : set R) =
        lebesgue_measure (`]a - 1, a[%classic%R : set R) +
        lebesgue_measure [set a])%E.
  rewrite lebesgue_measure_itvoo_subr1 lebesgue_measure_itvoc => /eqP.
  rewrite hlength_itv lte_fin ltr_subl_addr ltr_addl ltr01.
  rewrite [in X in X == _]/= EFinN EFinB oppeB// addeA subee// add0e.
  rewrite addeC -sube_eq//; last by rewrite fin_num_adde_def.
  by rewrite subee// => /eqP.
rewrite -setUitv1// ?bnd_simp; last by rewrite ltr_subl_addr ltr_addl.
rewrite measureU//; first exact: measurable_itv.
apply/seteqP; split => // x []/=; rewrite in_itv/= => + xa.
by rewrite xa ltxx andbF.
Qed.

Let lebesgue_measure_itvoo (a b : R) :
  (lebesgue_measure (`]a, b[ : set R) = hlength `]a, b[)%classic.
Proof.
have [ab|ba] := ltP a b; last by rewrite set_itv_ge ?measure0// -leNgt.
have := lebesgue_measure_itvoc a b.
rewrite 2!hlength_itv => <-; rewrite -setUitv1// measureU//.
- by have /= -> := lebesgue_measure_set1 b; rewrite adde0.
- exact: measurable_itv.
- by apply/seteqP; split => // x [/= + xb]; rewrite in_itv/= xb ltxx andbF.
Qed.

Let lebesgue_measure_itvcc (a b : R) :
  (lebesgue_measure (`[a, b] : set R) = hlength `[a, b])%classic.
Proof.
have [ab|ba] := leP a b; last by rewrite set_itv_ge ?measure0// -leNgt.
have := lebesgue_measure_itvoc a b.
rewrite 2!hlength_itv => <-; rewrite -setU1itv// measureU//.
- by have /= -> := lebesgue_measure_set1 a; rewrite add0e.
- exact: measurable_itv.
- by apply/seteqP; split => // x [/= ->]; rewrite in_itv/= ltxx.
Qed.

Let lebesgue_measure_itvco (a b : R) :
  (lebesgue_measure (`[a, b[ : set R) = hlength `[a, b[)%classic.
Proof.
have [ab|ba] := ltP a b; last by rewrite set_itv_ge ?measure0// -leNgt.
have := lebesgue_measure_itvoo a b.
rewrite 2!hlength_itv => <-; rewrite -setU1itv// measureU//.
- by have /= -> := lebesgue_measure_set1 a; rewrite add0e.
- exact: measurable_itv.
- by apply/seteqP; split => // x [/= ->]; rewrite in_itv/= ltxx.
Qed.

Let lebesgue_measure_itv_bnd (x y : bool) (a b : R) :
  lebesgue_measure ([set` Interval (BSide x a) (BSide y b)] : set R) =
  hlength [set` Interval (BSide x a) (BSide y b)].
Proof.
by move: x y => [|] [|]; [exact: lebesgue_measure_itvco |
  exact: lebesgue_measure_itvcc | exact: lebesgue_measure_itvoo |
  exact: lebesgue_measure_itvoc].
Qed.

Let limnatR : lim (fun k => (k%:R)%:E : \bar R) = +oo%E.
Proof.
apply/cvg_lim => //; apply/dvg_ereal_cvg/cvgPpinfty => A.
exists `|ceil A|%N => //= => n/=; rewrite -(@ler_nat R); apply: le_trans.
by rewrite natr_absz (le_trans (ceil_ge _))// intr_norm ler_norm.
Qed.

Let lebesgue_measure_itv_bnd_infty x (a : R) :
  lebesgue_measure ([set` Interval (BSide x a) +oo%O] : set R) = +oo%E.
Proof.
rewrite itv_bnd_infty_bigcup; transitivity (lim (lebesgue_measure \o
    (fun k => [set` Interval (BSide x a) (BRight (a + k%:R))] : set R))).
  apply/esym/cvg_lim => //; apply: cvg_mu_inc => //.
  + by move=> k; exact: measurable_itv.
  + by apply: bigcup_measurable => k _; exact: measurable_itv.
  + move=> m n mn; apply/subsetPset => r/=; rewrite !in_itv/= => /andP[->/=].
    by move=> /le_trans; apply; rewrite ler_add// ler_nat.
rewrite (_ : _ \o _ = (fun k => k%:R%:E))//.
apply/funext => n /=; rewrite lebesgue_measure_itv_bnd hlength_itv/=.
rewrite lte_fin;  have [->|n0] := eqVneq n 0%N; first by rewrite addr0 ltxx.
by rewrite ltr_addl ltr0n lt0n n0 EFinD addeAC EFinN subee ?add0e.
Qed.

Let lebesgue_measure_itv_infty_bnd y (b : R) :
  lebesgue_measure ([set` Interval -oo%O (BSide y b)] : set R) = +oo%E.
Proof.
rewrite itv_infty_bnd_bigcup; transitivity (lim (lebesgue_measure \o
    (fun k => [set` Interval (BLeft (b - k%:R)) (BSide y b)] : set R))).
  apply/esym/cvg_lim => //; apply: cvg_mu_inc => //.
  + by move=> k; exact: measurable_itv.
  + by apply: bigcup_measurable => k _; exact: measurable_itv.
  + move=> m n mn; apply/subsetPset => r/=; rewrite !in_itv/= => /andP[+ ->].
    by rewrite andbT; apply: le_trans; rewrite ler_sub// ler_nat.
rewrite (_ : _ \o _ = (fun k : nat => k%:R%:E))//.
apply/funext => n /=; rewrite lebesgue_measure_itv_bnd hlength_itv/= lte_fin.
have [->|n0] := eqVneq n 0%N; first by rewrite subr0 ltxx.
rewrite ltr_subl_addr ltr_addl ltr0n lt0n n0 EFinN EFinB oppeB// addeA subee//.
by rewrite add0e.
Qed.

Lemma lebesgue_measure_itv (i : interval R) :
  lebesgue_measure ([set` i] : set R) = hlength [set` i].
Proof.
move: i => [[x a|[|]]] [y b|[|]]; first exact: lebesgue_measure_itv_bnd.
- by rewrite set_itvE ?measure0.
- by rewrite lebesgue_measure_itv_bnd_infty hlength_bnd_infty.
- by rewrite lebesgue_measure_itv_infty_bnd hlength_infty_bnd.
- by rewrite set_itvE ?measure0.
- rewrite set_itvE hlength_setT.
  rewrite (_ : setT = [set` `]-oo, 0[] `|` [set` `[0, +oo[]); last first.
    by apply/seteqP; split=> // => x _; have [x0|x0] := leP 0 x; [right|left];
      rewrite /= in_itv//= x0.
  rewrite measureU//=; try exact: measurable_itv.
  + by rewrite lebesgue_measure_itv_infty_bnd lebesgue_measure_itv_bnd_infty.
  + by apply/seteqP; split => // x []/=; rewrite !in_itv/= andbT leNgt => ->.
- by rewrite set_itvE ?measure0.
- by rewrite set_itvE ?measure0.
- by rewrite set_itvE ?measure0.
Qed.

End lebesgue_measure_itv.

Lemma lebesgue_measure_rat (R : realType) :
  lebesgue_measure (range ratr : set R) = 0%E.
Proof.
have /pcard_eqP/bijPex[f bijf] := card_rat; set f1 := 'pinv_(fun=> 0) setT f.
rewrite (_ : range _ = \bigcup_n [set ratr (f1 n)]); last first.
  apply/seteqP; split => [_ [q _ <-]|_ [n _ /= ->]]; last by exists (f1 n).
  exists (f q) => //=; rewrite /f1 pinvKV// ?in_setE// => x y _ _.
  by apply: bij_inj; rewrite -setTT_bijective.
rewrite measure_bigcup//; last first.
  apply/trivIsetP => i j _ _ ij; apply/seteqP; split => //= _ [/= ->].
  move=> /fmorph_inj.
  have /set_bij_inj /[apply] := bijpinv_bij (fun=> 0) bijf.
  by rewrite in_setE => /(_ Logic.I Logic.I); exact/eqP.
by rewrite nneseries0// => n _; exact: lebesgue_measure_set1.
Qed.

Section measurable_fun_measurable.
Local Open Scope ereal_scope.
Variables (d : measure_display) (T : measurableType d).
Variables (R : realType) (D : set T) (f : T -> \bar R).
Hypotheses (mD : measurable D) (mf : measurable_fun D f).
Implicit Types y : \bar R.

Lemma emeasurable_fun_c_infty y : measurable (D `&` [set x | y <= f x]).
Proof.
by rewrite -preimage_itv_c_infty; exact/mf/emeasurable_itv_bnd_pinfty.
Qed.

Lemma emeasurable_fun_o_infty y :  measurable (D `&` [set x | y < f x]).
Proof.
by rewrite -preimage_itv_o_infty; exact/mf/emeasurable_itv_bnd_pinfty.
Qed.

Lemma emeasurable_fun_infty_o y : measurable (D `&` [set x | f x < y]).
Proof.
by rewrite -preimage_itv_infty_o; exact/mf/emeasurable_itv_ninfty_bnd.
Qed.

Lemma emeasurable_fun_infty_c y : measurable (D `&` [set x | f x <= y]).
Proof.
by rewrite -preimage_itv_infty_c; exact/mf/emeasurable_itv_ninfty_bnd.
Qed.

Lemma emeasurable_fin_num : measurable (D `&` [set x | f x \is a fin_num]).
Proof.
rewrite [X in measurable X](_ : _ =
  \bigcup_k (D `&` ([set  x | - k%:R%:E <= f x] `&` [set x | f x <= k%:R%:E]))).
  apply: bigcupT_measurable => k; rewrite -(setIid D) setIACA.
  by apply: measurableI; [exact: emeasurable_fun_c_infty|
                          exact: emeasurable_fun_infty_c].
rewrite predeqE => t; split => [/= [Dt ft]|].
  have [ft0|ft0] := leP 0%R (fine (f t)).
    exists `|ceil (fine (f t))|%N => //=; split => //; split.
      by rewrite -{2}(fineK ft)// lee_fin (le_trans _ ft0)// ler_oppl oppr0.
    by rewrite natr_absz ger0_norm ?ceil_ge0// -(fineK ft) lee_fin ceil_ge.
  exists `|floor (fine (f t))|%N => //=; split => //; split.
    rewrite natr_absz ltr0_norm ?floor_lt0// EFinN.
    by rewrite -{2}(fineK ft) lee_fin mulrNz opprK floor_le.
  by rewrite -(fineK ft)// lee_fin (le_trans (ltW ft0)).
move=> [n _] [/= Dt [nft fnt]]; split => //; rewrite fin_numElt.
by rewrite (lt_le_trans _ nft) ?ltNye//= (le_lt_trans fnt)// ltey.
Qed.

Lemma emeasurable_neq y : measurable (D `&` [set x | f x != y]).
Proof.
rewrite (_ : [set x | f x != y] = f @^-1` (setT `\ y)).
  exact/mf/measurableD.
rewrite predeqE => t; split; last by rewrite /preimage /= => -[_ /eqP].
by rewrite /= => ft0; rewrite /preimage /=; split => //; exact/eqP.
Qed.

End measurable_fun_measurable.

Module RGenOInfty.
Section rgenoinfty.
Variable R : realType.
Implicit Types x y z : R.

Definition G := [set A | exists x, A = `]x, +oo[%classic].

Lemma measurable_itv_bnd_infty b x :
  G.-sigma.-measurable [set` Interval (BSide b x) +oo%O].
Proof.
case: b; last by apply: sub_sigma_algebra; eexists; reflexivity.
rewrite itv_c_inftyEbigcap; apply: bigcapT_measurable => k.
by apply: sub_sigma_algebra; eexists; reflexivity.
Qed.

Lemma measurable_itv_bounded a b x : a != +oo%O ->
  G.-sigma.-measurable [set` Interval a (BSide b x)].
Proof.
case: a => [a r _|[_|//]].
  by rewrite set_itv_splitD; apply: measurableD => //;
    exact: measurable_itv_bnd_infty.
by rewrite -setCitvr; apply: measurableC; apply: measurable_itv_bnd_infty.
Qed.

Lemma measurableE :
  (R.-ocitv.-measurable).-sigma.-measurable = G.-sigma.-measurable.
Proof.
rewrite eqEsubset; split => A.
  apply: smallest_sub; first exact: smallest_sigma_algebra.
  by move=> I [x _ <-]; exact: measurable_itv_bounded.
apply: smallest_sub; first exact: smallest_sigma_algebra.
by move=> A' /= [x ->]; exact: measurable_itv.
Qed.

End rgenoinfty.
End RGenOInfty.

Module RGenInftyO.
Section rgeninftyo.
Variable R : realType.
Implicit Types x y z : R.

Definition G := [set A | exists x, A = `]-oo, x[%classic].

Lemma measurable_itv_bnd_infty b x :
  G.-sigma.-measurable [set` Interval -oo%O (BSide b x)].
Proof.
case: b; first by apply sub_sigma_algebra; eexists; reflexivity.
rewrite -setCitvr itv_o_inftyEbigcup; apply/measurableC/bigcupT_measurable => n.
rewrite -setCitvl; apply: measurableC.
by apply: sub_sigma_algebra; eexists; reflexivity.
Qed.

Lemma measurable_itv_bounded a b x : a != -oo%O ->
  G.-sigma.-measurable [set` Interval (BSide b x) a].
Proof.
case: a => [a r _|[//|_]].
  by rewrite set_itv_splitD; apply/measurableD => //;
     rewrite -setCitvl; apply: measurableC; exact: measurable_itv_bnd_infty.
by rewrite -setCitvl; apply: measurableC; apply: measurable_itv_bnd_infty.
Qed.

Lemma measurableE : (R.-ocitv.-measurable).-sigma.-measurable = G.-sigma.-measurable.
Proof.
rewrite eqEsubset; split => A.
  apply: smallest_sub; first exact: smallest_sigma_algebra.
  by move=> I [x _ <-]; apply: measurable_itv_bounded.
apply: smallest_sub; first exact: smallest_sigma_algebra.
by move=> A' /= [x ->]; apply: measurable_itv.
Qed.

End rgeninftyo.
End RGenInftyO.

Module RGenCInfty.
Section rgencinfty.
Variable R : realType.
Implicit Types x y z : R.

Definition G : set (set R) := [set A | exists x, A = `[x, +oo[%classic].

Lemma measurable_itv_bnd_infty b x :
  G.-sigma.-measurable [set` Interval (BSide b x) +oo%O].
Proof.
case: b; first by apply: sub_sigma_algebra; exists x; rewrite set_itv_c_infty.
rewrite itv_o_inftyEbigcup; apply: bigcupT_measurable => k.
by apply: sub_sigma_algebra; eexists; reflexivity.
Qed.

Lemma measurable_itv_bounded a b y : a != +oo%O ->
  G.-sigma.-measurable [set` Interval a (BSide b y)].
Proof.
case: a => [a r _|[_|//]].
  rewrite set_itv_splitD.
  by apply: measurableD; apply: measurable_itv_bnd_infty.
by rewrite -setCitvr; apply: measurableC; apply: measurable_itv_bnd_infty.
Qed.

Lemma measurableE : (R.-ocitv.-measurable).-sigma.-measurable = G.-sigma.-measurable.
Proof.
rewrite eqEsubset; split => A.
  apply: smallest_sub; first exact: smallest_sigma_algebra.
  by move=> I [x _ <-]; apply: measurable_itv_bounded.
apply: smallest_sub; first exact: smallest_sigma_algebra.
by move=> A' /= [x ->]; apply: measurable_itv.
Qed.

End rgencinfty.
End RGenCInfty.

Module RGenOpens.
Section rgenopens.

Variable R : realType.
Implicit Types x y z : R.

Definition G := [set A | exists x y, A = `]x, y[%classic].

Local Lemma measurable_itvoo x y : G.-sigma.-measurable `]x, y[%classic.
Proof. by apply sub_sigma_algebra; eexists; eexists; reflexivity. Qed.

Local Lemma measurable_itv_o_infty x : G.-sigma.-measurable `]x, +oo[%classic.
Proof.
rewrite itv_bnd_inftyEbigcup; apply: bigcupT_measurable => i.
exact: measurable_itvoo.
Qed.

Lemma measurable_itv_bnd_infty b x :
  G.-sigma.-measurable [set` Interval (BSide b x) +oo%O].
Proof.
case: b; last exact: measurable_itv_o_infty.
rewrite itv_c_inftyEbigcap; apply: bigcapT_measurable => k.
exact: measurable_itv_o_infty.
Qed.

Lemma measurable_itv_infty_bnd b x :
  G.-sigma.-measurable [set` Interval -oo%O (BSide b x)].
Proof.
by rewrite -setCitvr; apply: measurableC; exact: measurable_itv_bnd_infty.
Qed.

Lemma measurable_itv_bounded a x b y :
  G.-sigma.-measurable [set` Interval (BSide a x) (BSide b y)].
Proof.
move: a b => [] []; rewrite -[X in measurable X]setCK setCitv;
  apply: measurableC; apply: measurableU; try solve[
    exact: measurable_itv_infty_bnd|exact: measurable_itv_bnd_infty].
Qed.

Lemma measurableE : (R.-ocitv.-measurable).-sigma.-measurable = G.-sigma.-measurable.
Proof.
rewrite eqEsubset; split => A.
  apply: smallest_sub; first exact: smallest_sigma_algebra.
  by move=> I [x _ <-]; apply: measurable_itv_bounded.
apply: smallest_sub; first exact: smallest_sigma_algebra.
by move=> A' /= [x [y ->]]; apply: measurable_itv.
Qed.

End rgenopens.
End RGenOpens.

Section erealwithrays.
Variable R : realType.
Implicit Types (x y z : \bar R) (r s : R).
Local Open Scope ereal_scope.

Lemma EFin_itv_bnd_infty b r : EFin @` [set` Interval (BSide b r) +oo%O] =
  [set` Interval (BSide b r%:E) +oo%O] `\ +oo.
Proof.
rewrite eqEsubset; split => [x [s /itvP rs <-]|x []].
  split => //=; rewrite in_itv /=.
  by case: b in rs *; rewrite /= ?(lee_fin, lte_fin) rs.
move: x => [s|_ /(_ erefl)|] //=; rewrite in_itv /= andbT; last first.
  by case: b => /=; rewrite 1?(leNgt,ltNge) 1?(ltNye, leNye).
by case: b => /=; rewrite 1?(lte_fin,lee_fin) => rs _;
  exists s => //; rewrite in_itv /= rs.
Qed.

Lemma EFin_itv r : [set s | r%:E < s%:E] = `]r, +oo[%classic.
Proof.
by rewrite predeqE => s; split => [|]; rewrite /= lte_fin in_itv/= andbT.
Qed.

Lemma preimage_EFin_setT : @EFin R @^-1` [set x | x \in `]-oo%E, +oo[] = setT.
Proof.
by rewrite set_itvE predeqE => r; split=> // _; rewrite /preimage /= ltNye.
Qed.

Lemma eitv_c_infty r : `[r%:E, +oo[%classic =
  \bigcap_k `](r - k.+1%:R^-1)%:E, +oo[%classic :> set _.
Proof.
rewrite predeqE => x; split=> [|].
- move: x => [s /=| _ n _|//].
  + rewrite in_itv /= andbT lee_fin => rs n _ /=.
    by rewrite in_itv/= andbT lte_fin ltr_subl_addl (le_lt_trans rs)// ltr_addr.
  + by rewrite /= in_itv /= andbT ltey.
- move: x => [s| |/(_ 0%N Logic.I)] //=; last by rewrite in_itv /= leey.
  move=> h; rewrite in_itv /= lee_fin leNgt andbT; apply/negP.
  move=> /ltr_add_invr[k skr]; have {h} := h k Logic.I.
  rewrite /= in_itv /= andbT lte_fin ltNge => /negP; apply.
  by rewrite -ler_subl_addr opprK ltW.
Qed.

Lemma eitv_infty_c r : `]-oo, r%:E]%classic =
  \bigcap_k `]-oo, (r%:E + k.+1%:R^-1%:E)]%classic :> set _.
Proof.
rewrite predeqE => x; split=> [|].
- move: x => [s /=|//|_ n _].
  + rewrite in_itv /= lee_fin => sr n _; rewrite /= in_itv /=.
    by rewrite -EFinD lee_fin (le_trans sr)// ler_addl.
  + by rewrite /= in_itv /= -EFinD leNye.
- move: x => [s|/(_ 0%N Logic.I)//|]/=; rewrite ?in_itv /= ?leNye//.
  move=> h; rewrite lee_fin leNgt; apply/negP => /ltr_add_invr[k rks].
  have {h} := h k Logic.I; rewrite /= in_itv /=.
  by rewrite -EFinD lee_fin leNgt => /negP; apply.
Qed.

Lemma eset1_ninfty :
  [set -oo] = \bigcap_k `]-oo, (-k%:R%:E)[%classic :> set (\bar R).
Proof.
rewrite eqEsubset; split=> [_ -> i _ |]; first by rewrite /= in_itv /= ltNye.
move=> [r|/(_ O Logic.I)|]//.
move=> /(_ `|floor r|%N Logic.I); rewrite /= in_itv/= ltNge.
rewrite lee_fin; have [r0|r0] := leP 0%R r.
  by rewrite (le_trans _ r0) // ler_oppl oppr0 ler0n.
rewrite ler_oppl -abszN natr_absz gtr0_norm; last first.
  by rewrite ltr_oppr oppr0 floor_lt0.
by rewrite mulrNz ler_oppl opprK floor_le.
Qed.

Lemma eset1_pinfty :
  [set +oo] = \bigcap_k `]k%:R%:E, +oo[%classic :> set (\bar R).
Proof.
rewrite eqEsubset; split=> [_ -> i _/=|]; first by rewrite in_itv /= ltey.
move=> [r| |/(_ O Logic.I)] // /(_ `|ceil r|%N Logic.I); rewrite /= in_itv /=.
rewrite andbT lte_fin ltNge.
have [r0|r0] := ltP 0%R r; last by rewrite (le_trans r0).
by rewrite natr_absz gtr0_norm // ?ceil_ge// ceil_gt0.
Qed.

End erealwithrays.

Module ErealGenOInfty.
Section erealgenoinfty.
Variable R : realType.
Implicit Types (x y z : \bar R) (r s : R).

Local Open Scope ereal_scope.

Definition G := [set A : set \bar R | exists x, A = `]x, +oo[%classic].

Lemma measurable_set1_ninfty : G.-sigma.-measurable [set -oo].
Proof.
rewrite eset1_ninfty; apply: bigcap_measurable => i _.
rewrite -setCitvr; apply: measurableC; rewrite eitv_c_infty.
apply: bigcap_measurable => j _; apply: sub_sigma_algebra.
by exists (- (i%:R + j.+1%:R^-1))%:E; rewrite opprD.
Qed.

Lemma measurable_set1_pinfty : G.-sigma.-measurable [set +oo].
Proof.
rewrite eset1_pinfty; apply: bigcapT_measurable => i.
by apply: sub_sigma_algebra; exists i%:R%:E.
Qed.

Lemma measurableE : emeasurable (R.-ocitv.-measurable) = G.-sigma.-measurable.
Proof.
apply/seteqP; split; last first.
  apply: smallest_sub.
    split; first exact: emeasurable0.
      by move=> *; rewrite setTD; exact: emeasurableC.
    by move=> *; exact: bigcupT_emeasurable.
  move=> _ [x ->]; rewrite /emeasurable /=; move: x => [r| |].
  + exists `]r, +oo[%classic.
      rewrite RGenOInfty.measurableE.
      exact: RGenOInfty.measurable_itv_bnd_infty.
    by exists [set +oo]; [constructor|rewrite -punct_eitv_bnd_pinfty].
  + exists set0 => //.
    by exists set0; [constructor|rewrite setU0 itv_opinfty_pinfty image_set0].
  + exists setT => //; exists [set +oo]; first by constructor.
    by rewrite itv_oninfty_pinfty punct_eitv_setTR.
move=> A [B mB [C mC]] <-; apply: measurableU; last first.
  case: mC; [by []|exact: measurable_set1_ninfty
                  |exact: measurable_set1_pinfty|].
  - by apply: measurableU; [exact: measurable_set1_ninfty|
                            exact: measurable_set1_pinfty].
rewrite RGenOInfty.measurableE in mB.
have smB := smallest_sub _ _ mB.
(* BUG: elim/smB : _. fails !! *)
apply: (smB (G.-sigma.-measurable \o (image^~ EFin))); last first.
  move=> _ [r ->]/=; rewrite EFin_itv_bnd_infty; apply: measurableD.
    by apply sub_sigma_algebra => /=; exists r%:E.
  exact: measurable_set1_pinfty.
split=> /= [|D mD|F mF]; first by rewrite image_set0.
- rewrite setTD EFin_setC; apply: measurableD; first exact: measurableC.
  by apply: measurableU; [exact: measurable_set1_ninfty|
                          exact: measurable_set1_pinfty].
- by rewrite EFin_bigcup; apply: bigcup_measurable => i _ ; exact: mF.
Qed.

End erealgenoinfty.
End ErealGenOInfty.

Module ErealGenCInfty.
Section erealgencinfty.
Variable R : realType.
Implicit Types (x y z : \bar R) (r s : R).
Local Open Scope ereal_scope.

Definition G := [set A : set \bar R | exists x, A = `[x, +oo[%classic].

Lemma measurable_set1_ninfty : G.-sigma.-measurable [set -oo].
Proof.
rewrite eset1_ninfty; apply: bigcapT_measurable=> i; rewrite -setCitvr.
by apply: measurableC; apply: sub_sigma_algebra; exists (- i%:R)%:E.
Qed.

Lemma measurable_set1_pinfty : G.-sigma.-measurable [set +oo].
Proof.
apply: sub_sigma_algebra; exists +oo; rewrite predeqE => x; split => [->//|/=].
by rewrite in_itv /= andbT leye_eq => /eqP ->.
Qed.

Lemma measurableE : emeasurable (R.-ocitv.-measurable) = G.-sigma.-measurable.
Proof.
apply/seteqP; split; last first.
  apply: smallest_sub.
    split; first exact: emeasurable0.
      by move=> *; rewrite setTD; exact: emeasurableC.
    by move=> *; exact: bigcupT_emeasurable.
  move=> _ [[r||] ->]/=.
  - exists `[r, +oo[%classic.
      rewrite RGenOInfty.measurableE.
      exact: RGenOInfty.measurable_itv_bnd_infty.
    by exists [set +oo]; [constructor | rewrite -punct_eitv_bnd_pinfty].
   - exists set0 => //; exists [set +oo]; first by constructor.
     by rewrite image_set0 set0U itv_cpinfty_pinfty.
   - exists setT => //; exists [set -oo; +oo]; first by constructor.
     by rewrite itv_cninfty_pinfty setUA punct_eitv_setTL setUCl.
move=> _ [A' mA' [C mC]] <-; apply: measurableU; last first.
  case: mC; [by []|exact: measurable_set1_ninfty|
                   exact: measurable_set1_pinfty|].
  by apply: measurableU; [exact: measurable_set1_ninfty|
                          exact: measurable_set1_pinfty].
rewrite RGenCInfty.measurableE in mA'.
have smA' := smallest_sub _ _ mA'.
(* BUG: elim/smA' : _. fails !! *)
apply: (smA' (G.-sigma.-measurable \o (image^~ EFin))); last first.
  move=> _ [r ->]/=; rewrite EFin_itv_bnd_infty; apply: measurableD.
    by apply sub_sigma_algebra => /=; exists r%:E.
  exact: measurable_set1_pinfty.
split=> /= [|D mD|F mF]; first by rewrite image_set0.
- rewrite setTD EFin_setC; apply: measurableD; first exact: measurableC.
  by apply: measurableU; [exact: measurable_set1_ninfty|
                          exact: measurable_set1_pinfty].
- by rewrite EFin_bigcup; apply: bigcup_measurable => i _; exact: mF.
Qed.

End erealgencinfty.
End ErealGenCInfty.

Section trace.
Variable (T : Type).
Implicit Types (G : set (set T)) (A D : set T).

(* intended as a trace sigma-algebra *)
Definition strace G D := [set x `&` D | x in G].

Lemma stracexx G D : G D -> strace G D D.
Proof. by rewrite /strace /=; exists D => //; rewrite setIid. Qed.

Lemma sigma_algebra_strace G D :
  sigma_algebra setT G -> sigma_algebra D (strace G D).
Proof.
move=> [G0 GC GU]; split; first by exists set0 => //; rewrite set0I.
- move=> S [A mA ADS]; have mCA := GC _ mA.
  have : strace G D (D `&` ~` A).
    by rewrite setIC; exists (setT `\` A) => //; rewrite setTD.
  rewrite -setDE => trDA.
  have DADS : D `\` A = D `\` S by rewrite -ADS !setDE setCI setIUr setICr setU0.
  by rewrite DADS in trDA.
- move=> S mS; have /choice[M GM] : forall n, exists A, G A /\ S n = A `&` D.
    by move=> n; have [A mA ADSn] := mS n; exists A.
  exists (\bigcup_i (M i)); first by apply GU => i;  exact: (GM i).1.
  by rewrite setI_bigcupl; apply eq_bigcupr => i _; rewrite (GM i).2.
Qed.

End trace.

Lemma strace_measurable d (T : measurableType d) (A : set T) : measurable A ->
  strace measurable A `<=` measurable.
Proof. by move=> mA=> _ [C mC <-]; apply: measurableI. Qed.

(* more properties of measurable functions *)

Lemma is_interval_measurable (R : realType) (I : set R) :
  is_interval I -> measurable I.
Proof. by move/is_intervalP => ->; exact: measurable_itv. Qed.

Section coutinuous_measurable.
Variable R : realType.

Lemma open_measurable (U : set R) : open U -> measurable U.
Proof.
move=> /open_bigcup_rat ->; rewrite bigcup_mkcond; apply: bigcupT_measurable_rat.
move=> q; case: ifPn => // qfab; apply: is_interval_measurable => //.
exact: is_interval_bigcup_ointsub.
Qed.

Lemma continuous_measurable_fun (f : R -> R) : continuous f ->
  measurable_fun setT f.
Proof.
move=> /continuousP cf; apply: (measurability (RGenOpens.measurableE R)).
move=> _ [_ [a [b ->] <-]]; rewrite setTI.
by apply: open_measurable; exact/cf/interval_open.
Qed.

End coutinuous_measurable.

Section standard_measurable_fun.

Lemma measurable_fun_normr (R : realType) (D : set R) :
  measurable_fun D (@normr _ R).
Proof.
move=> mD; apply: (measurability (RGenOInfty.measurableE R)) => //.
move=> /= _ [_ [x ->] <-]; apply: measurableI => //.
have [x0|x0] := leP 0 x.
  rewrite [X in measurable X](_ : _ = `]-oo, (- x)[ `|` `]x, +oo[)%classic.
    by apply: measurableU; apply: measurable_itv.
  rewrite predeqE => r; split => [|[|]]; rewrite preimage_itv ?in_itv ?andbT/=.
  - have [r0|r0] := leP 0 r; [rewrite ger0_norm|rewrite ltr0_norm] => // xr;
      rewrite 2!in_itv/=.
    + by right; rewrite xr.
    + by left; rewrite ltr_oppr.
  - move=> rx /=.
    by rewrite ler0_norm 1?ltr_oppr// (le_trans (ltW rx))// ler_oppl oppr0.
  - by rewrite in_itv /= andbT => xr; rewrite (lt_le_trans _ (ler_norm _)).
rewrite [X in measurable X](_ : _ = setT)// predeqE => r.
by split => // _; rewrite /= in_itv /= andbT (lt_le_trans x0).
Qed.

End standard_measurable_fun.

Section measurable_fun_realType.
Variables (d : measure_display) (T : measurableType d) (R : realType).
Implicit Types (D : set T) (f g : T -> R).

Lemma measurable_funD D f g :
  measurable_fun D f -> measurable_fun D g -> measurable_fun D (f \+ g).
Proof.
move=> mf mg mD; apply: (measurability (RGenOInfty.measurableE R)) => //.
move=> /= _ [_ [a ->] <-]; rewrite preimage_itv_o_infty.
rewrite [X in measurable X](_ : _ = \bigcup_(q : rat)
  ((D `&` [set x | ratr q < f x]) `&` (D `&` [set x | a - ratr q < g x]))).
  apply: bigcupT_measurable_rat => q; apply: measurableI.
  - by rewrite -preimage_itv_o_infty; apply: mf => //; apply: measurable_itv.
  - by rewrite -preimage_itv_o_infty; apply: mg => //; apply: measurable_itv.
rewrite predeqE => x; split => [|[r _] []/= [Dx rfx]] /= => [[Dx]|[_]].
  rewrite -ltr_subl_addr => /rat_in_itvoo[r]; rewrite inE /= => /itvP h.
  exists r => //; rewrite setIACA setIid; split => //; split => /=.
    by rewrite h.
  by rewrite ltr_subl_addr addrC -ltr_subl_addr h.
by rewrite ltr_subl_addr=> afg; rewrite (lt_le_trans afg)// addrC ler_add2r ltW.
Qed.

Lemma measurable_funrM D f (k : R) : measurable_fun D f ->
  measurable_fun D (fun x => k * f x).
Proof.
apply: (@measurable_fun_comp _ _ _ _ _ _ ( *%R k)).
by apply: continuous_measurable_fun; apply: mulrl_continuous.
Qed.

Lemma measurable_funN D f : measurable_fun D f -> measurable_fun D (-%R \o f).
Proof.
move=> mf mD; rewrite (_ : _ \o _ = (fun x => - 1 * f x)).
  exact: measurable_funrM.
by under eq_fun do rewrite mulN1r.
Qed.

Lemma measurable_funB D f g : measurable_fun D f ->
  measurable_fun D g -> measurable_fun D (f \- g).
Proof.
by move=> ? ? ?; apply: measurable_funD => //; exact: measurable_funN.
Qed.

Lemma measurable_fun_exprn D n f :
  measurable_fun D f -> measurable_fun D (fun x => f x ^+ n).
Proof.
apply: measurable_fun_comp ((@GRing.exp R)^~ n) _ _ _.
by apply: continuous_measurable_fun; apply: exprn_continuous.
Qed.

Lemma measurable_fun_sqr D f :
  measurable_fun D f -> measurable_fun D (fun x => f x ^+ 2).
Proof. exact: measurable_fun_exprn. Qed.

Lemma measurable_funM D f g :
  measurable_fun D f -> measurable_fun D g -> measurable_fun D (f \* g).
Proof.
move=> mf mg mD; rewrite (_ : (_ \* _) = (fun x => 2%:R^-1 * (f x + g x) ^+ 2)
  \- (fun x => 2%:R^-1 * (f x ^+ 2)) \- (fun x => 2%:R^-1 * ( g x ^+ 2))).
  apply: measurable_funB => //; last first.
    by apply: measurable_funrM => //; exact: measurable_fun_sqr.
  apply: measurable_funB => //; last first.
    by apply: measurable_funrM => //; exact: measurable_fun_sqr.
  apply: measurable_funrM => //.
  by apply: measurable_fun_sqr => //; exact: measurable_funD.
rewrite funeqE => x /=; rewrite -2!mulrBr sqrrD (addrC (f x ^+ 2)) -addrA.
rewrite -(addrA (f x * g x *+ 2)) -opprB opprK (addrC (g x ^+ 2)) addrK.
by rewrite -(mulr_natr (f x * g x)) -(mulrC 2) mulrA mulVr ?mul1r// unitfE.
Qed.

Lemma measurable_fun_max  D f g :
  measurable_fun D f -> measurable_fun D g -> measurable_fun D (f \max g).
Proof.
move=> mf mg mD; apply (measurability (RGenCInfty.measurableE R)) => //.
move=> _ [_ [x ->] <-]; rewrite [X in measurable X](_ : _ =
    (D `&` f @^-1` `[x, +oo[) `|` (D `&` g @^-1` `[x, +oo[)); last first.
  rewrite predeqE => t /=; split.
    by rewrite /= !in_itv /= !andbT le_maxr => -[Dx /orP[|]]; tauto.
  by move=> [|]; rewrite !in_itv/= !andbT le_maxr => -[Dx ->]//; rewrite orbT.
by apply: measurableU; [apply: mf|apply: mg] =>//; apply: measurable_itv.
Qed.

Lemma measurable_fun_sups D (h : (T -> R)^nat) n :
  (forall t, D t -> has_ubound (range (h ^~ t))) ->
  (forall m, measurable_fun D (h m)) ->
  measurable_fun D (fun x => sups (h ^~ x) n).
Proof.
move=> f_ub mf mD; apply: (measurability (RGenOInfty.measurableE R)) => //.
move=> _ [_ [x ->] <-]; rewrite sups_preimage // setI_bigcupr.
by apply: bigcup_measurable => k /= nk; apply: mf => //; exact: measurable_itv.
Qed.

Lemma measurable_fun_infs D (h : (T -> R)^nat) n :
  (forall t, D t -> has_lbound (range (h ^~ t))) ->
  (forall n, measurable_fun D (h n)) ->
  measurable_fun D (fun x => infs (h ^~ x) n).
Proof.
move=> lb_f mf mD; apply: (measurability (RGenInftyO.measurableE R)) =>//.
move=> _ [_ [x ->] <-]; rewrite infs_preimage // setI_bigcupr.
by apply: bigcup_measurable => k /= nk; apply: mf => //; exact: measurable_itv.
Qed.

Lemma measurable_fun_lim_sup D (h : (T -> R)^nat) :
  (forall t, D t -> has_ubound (range (h ^~ t))) ->
  (forall t, D t -> has_lbound (range (h ^~ t))) ->
  (forall n, measurable_fun D (h n)) ->
  measurable_fun D (fun x => lim_sup (h ^~ x)).
Proof.
move=> f_ub f_lb mf.
have : {in D, (fun x => inf [set sups (h ^~ x) n | n in [set n | 0 <= n]%N])
              =1 (fun x => lim_sup (h^~ x))}.
  move=> t; rewrite inE => Dt; apply/esym/cvg_lim; first exact: Rhausdorff.
  rewrite [X in _ --> X](_ : _ = inf (range (sups (h^~t)))).
    by apply: cvg_sups_inf; [exact: f_ub|exact: f_lb].
  by congr (inf [set _ | _ in _]); rewrite predeqE.
move/eq_measurable_fun; apply; apply: measurable_fun_infs => //.
  move=> t Dt; have [M hM] := f_lb _ Dt; exists M => _ [m /= nm <-].
  rewrite (@le_trans _ _ (h m t)) //; first by apply hM => /=; exists m.
  by apply: sup_ub; [exact/has_ubound_sdrop/f_ub|exists m => /=].
by move=> k; exact: measurable_fun_sups.
Qed.

Lemma measurable_fun_cvg D (h : (T -> R)^nat) f :
  (forall m, measurable_fun D (h m)) -> (forall x, D x -> h ^~ x --> f x) ->
  measurable_fun D f.
Proof.
move=> mf_ f_f; have fE x : D x -> f x = lim_sup (h ^~ x).
  move=> Dx; have /cvg_lim  <-// := @cvg_sups _ (h ^~ x) (f x) (f_f _ Dx).
  exact: Rhausdorff.
apply: (@eq_measurable_fun _ _ _ _ D (fun x => lim_sup (h ^~ x))).
  by move=> x; rewrite inE => Dx; rewrite -fE.
apply: (@measurable_fun_lim_sup _ h) => // t Dt.
- apply/bounded_fun_has_ubound/(@cvg_seq_bounded _ [normedModType R of R^o]).
  by apply/cvg_ex; eexists; exact: f_f.
- apply/bounded_fun_has_lbound/(@cvg_seq_bounded _ [normedModType R of R^o]).
  by apply/cvg_ex; eexists; exact: f_f.
Qed.

End measurable_fun_realType.

Section standard_emeasurable_fun.
Variable R : realType.

Lemma measurable_fun_EFin (D : set R) : measurable_fun D EFin.
Proof.
move=> mD; apply: (measurability (ErealGenOInfty.measurableE R)) => //.
move=> /= _ [_ [x ->]] <-; move: x => [x| |]; apply: measurableI => //.
- by rewrite preimage_itv_o_infty EFin_itv; exact: measurable_itv.
- by rewrite [X in measurable X](_ : _ = set0)// predeqE.
- by rewrite preimage_EFin_setT.
Qed.

Lemma measurable_fun_abse (D : set (\bar R)) : measurable_fun D abse.
Proof.
move=> mD; apply: (measurability (ErealGenOInfty.measurableE R)) => //.
move=> /= _ [_ [x ->] <-]; move: x => [x| |].
- rewrite [X in _ @^-1` X](punct_eitv_bnd_pinfty _ x) preimage_setU setIUr.
  apply: measurableU; last first.
    rewrite preimage_abse_pinfty.
    by apply: measurableI => //; exact: measurableU.
  apply: measurableI => //; exists (normr @^-1` `]x, +oo[%classic).
    rewrite -[X in measurable X]setTI.
    by apply: measurable_fun_normr => //; exact: measurable_itv.
  exists set0; first by constructor.
  rewrite setU0 predeqE => -[y| |]; split => /= => -[r];
    rewrite ?/= /= ?in_itv /= ?andbT => xr//.
    + by move=> [ry]; exists `|y| => //=; rewrite in_itv/= andbT -ry.
    + by move=> [ry]; exists y => //=; rewrite /= in_itv/= andbT -ry.
- by apply: measurableI => //; rewrite itv_opinfty_pinfty preimage_set0.
- apply: measurableI => //; rewrite itv_oninfty_pinfty -preimage_setC.
  by apply: measurableC; rewrite preimage_abse_ninfty.
Qed.

Lemma emeasurable_fun_minus (D : set (\bar R)) :
  measurable_fun D (-%E : \bar R -> \bar R).
Proof.
move=> mD; apply: (measurability (ErealGenCInfty.measurableE R)) => //.
move=> _ [_ [x ->] <-]; rewrite (_ : _ @^-1` _ = `]-oo, (- x)%E]%classic).
  by apply: measurableI => //; exact: emeasurable_itv_ninfty_bnd.
by rewrite predeqE => y; rewrite preimage_itv !in_itv/= andbT in_itv lee_oppr.
Qed.

End standard_emeasurable_fun.
#[global] Hint Extern 0 (measurable_fun _ abse) =>
  solve [exact: measurable_fun_abse] : core.
#[global] Hint Extern 0 (measurable_fun _ EFin) =>
  solve [exact: measurable_fun_EFin] : core.

(* NB: real-valued function *)
Lemma EFin_measurable_fun d (T : measurableType d) (R : realType) (D : set T)
    (g : T -> R) :
  measurable_fun D (EFin \o g) <-> measurable_fun D g.
Proof.
split=> [mf mD A mA|]; last by move=> mg; exact: measurable_fun_comp.
rewrite [X in measurable X](_ : _ = D `&` (EFin \o g) @^-1` (EFin @` A)).
  by apply: mf => //; exists A => //; exists set0; [constructor|rewrite setU0].
congr (_ `&` _);rewrite eqEsubset; split=> [|? []/= _ /[swap] -[->//]].
by move=> ? ?; exact: preimage_image.
Qed.

Section emeasurable_fun.
Local Open Scope ereal_scope.
Variables (d : measure_display) (T : measurableType d) (R : realType).
Implicit Types (D : set T).

Lemma measurable_fun_einfs D (f : (T -> \bar R)^nat) :
  (forall n, measurable_fun D (f n)) ->
  forall n, measurable_fun D (fun x => einfs (f ^~ x) n).
Proof.
move=> mf n mD.
apply: (measurability (ErealGenCInfty.measurableE R)) => //.
move=> _ [_ [x ->] <-]; rewrite einfs_preimage -bigcapIr; last by exists n => /=.
by apply: bigcap_measurable => ? ?; exact/mf/emeasurable_itv_bnd_pinfty.
Qed.

Lemma measurable_fun_esups D (f : (T -> \bar R)^nat) :
  (forall n, measurable_fun D (f n)) ->
  forall n, measurable_fun D (fun x => esups (f ^~ x) n).
Proof.
move=> mf n mD; apply: (measurability (ErealGenOInfty.measurableE R)) => //.
move=> _ [_ [x ->] <-];rewrite esups_preimage setI_bigcupr.
by apply: bigcup_measurable => ? ?; exact/mf/emeasurable_itv_bnd_pinfty.
Qed.

Lemma emeasurable_fun_max D (f g : T -> \bar R) :
  measurable_fun D f -> measurable_fun D g ->
  measurable_fun D (fun x => maxe (f x) (g x)).
Proof.
move=> mf mg mD; apply: (measurability (ErealGenCInfty.measurableE R)) => //.
move=> _ [_ [x ->] <-]; rewrite [X in measurable X](_ : _ =
    (D `&` f @^-1` `[x, +oo[) `|` (D `&` g @^-1` `[x, +oo[)); last first.
  rewrite predeqE => t /=; split.
    by rewrite !/= /= !in_itv /= !andbT le_maxr => -[Dx /orP[|]];
      tauto.
  by move=> [|]; rewrite !/= /= !in_itv/= !andbT le_maxr;
    move=> [Dx ->]//; rewrite orbT.
by apply: measurableU; [exact/mf/emeasurable_itv_bnd_pinfty|
                        exact/mg/emeasurable_itv_bnd_pinfty].
Qed.

Lemma emeasurable_funN D (f : T -> \bar R) :
  measurable_fun D f -> measurable_fun D (\- f).
Proof. by apply: measurable_fun_comp => //; exact: emeasurable_fun_minus. Qed.

Lemma emeasurable_fun_funepos D (f : T -> \bar R) :
  measurable_fun D f -> measurable_fun D f^\+.
Proof.
by move=> mf; apply: emeasurable_fun_max => //; exact: measurable_fun_cst.
Qed.

Lemma emeasurable_fun_funeneg D (f : T -> \bar R) :
  measurable_fun D f -> measurable_fun D f^\-.
Proof.
by move=> mf; apply: emeasurable_fun_max => //;
  [exact: emeasurable_funN|exact: measurable_fun_cst].
Qed.

Lemma emeasurable_fun_min D (f g : T -> \bar R) :
  measurable_fun D f -> measurable_fun D g ->
  measurable_fun D (fun x => mine (f x) (g x)).
Proof.
move=> /emeasurable_funN mf /emeasurable_funN mg.
have /emeasurable_funN := emeasurable_fun_max mf mg.
by apply eq_measurable_fun => i Di; rewrite -oppe_min oppeK.
Qed.

Lemma measurable_fun_elim_sup D (f : (T -> \bar R)^nat) :
  (forall n, measurable_fun D (f n)) ->
  measurable_fun D (fun x => elim_sup (f ^~ x)).
Proof.
move=> mf mD; rewrite (_ :  (fun _ => _) =
    (fun x => ereal_inf [set esups (f^~ x) n | n in [set n | n >= 0]%N])).
  by apply: measurable_fun_einfs => // k; exact: measurable_fun_esups.
rewrite funeqE => t; apply/cvg_lim => //.
rewrite [X in _ --> X](_ : _ = ereal_inf (range (esups (f^~t)))).
  exact: cvg_esups_inf.
by congr (ereal_inf [set _ | _ in _]); rewrite predeqE.
Qed.

Lemma emeasurable_fun_cvg D (f_ : (T -> \bar R)^nat) (f : T -> \bar R) :
  (forall m, measurable_fun D (f_ m)) ->
  (forall x, D x -> f_ ^~ x --> f x) -> measurable_fun D f.
Proof.
move=> mf_ f_f; have fE x : D x -> f x = elim_sup (f_^~ x).
  by move=> Dx; have /cvg_lim  <-// := @cvg_esups _ (f_^~x) (f x) (f_f x Dx).
apply: (measurable_fun_ext (fun x => elim_sup (f_ ^~ x))) => //.
  by move=> x; rewrite inE => Dx; rewrite fE.
exact: measurable_fun_elim_sup.
Qed.

End emeasurable_fun.
Arguments emeasurable_fun_cvg {d T R D} f_.