File: Mathlib/MeasureTheory/Integral/SetIntegral.lean

full_name: MeasureTheory.setIntegral_congr_ae₀
code: theorem setIntegral_congr_ae₀ (hs : NullMeasurableSet s μ) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
    ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, g x ∂μ

full_name: MeasureTheory.setIntegral_congr_ae
code: theorem setIntegral_congr_ae (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x = g x) :
    ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, g x ∂μ

full_name: MeasureTheory.setIntegral_congr₀
code: theorem setIntegral_congr₀ (hs : NullMeasurableSet s μ) (h : EqOn f g s) :
    ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, g x ∂μ

full_name: MeasureTheory.setIntegral_congr
code: theorem setIntegral_congr (hs : MeasurableSet s) (h : EqOn f g s) :
    ∫ x in s, f x ∂μ = ∫ x in s, g x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, g x ∂μ

full_name: MeasureTheory.setIntegral_congr_set_ae
code: theorem setIntegral_congr_set_ae (hst : s =ᵐ[μ] t) : ∫ x in s, f x ∂μ = ∫ x in t, f x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in t, f x ∂μ

full_name: MeasureTheory.integral_union_ae
code: theorem integral_union_ae (hst : AEDisjoint μ s t) (ht : NullMeasurableSet t μ)
    (hfs : IntegrableOn f s μ) (hft : IntegrableOn f t μ) :
    ∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ + ∫ x in t, f x ∂μ 
goal: ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in t, f x ∂μ

full_name: MeasureTheory.integral_union
code: theorem integral_union (hst : Disjoint s t) (ht : MeasurableSet t) (hfs : IntegrableOn f s μ)
    (hft : IntegrableOn f t μ) : ∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ + ∫ x in t, f x ∂μ 
goal: ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in t, f x ∂μ

full_name: MeasureTheory.integral_diff
code: theorem integral_diff (ht : MeasurableSet t) (hfs : IntegrableOn f s μ) (hts : t ⊆ s) :
    ∫ x in s \ t, f x ∂μ = ∫ x in s, f x ∂μ - ∫ x in t, f x ∂μ 
goal: ∫ (x : X) in s \ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ - ∫ (x : X) in t, f x ∂μ

full_name: MeasureTheory.integral_inter_add_diff₀
code: theorem integral_inter_add_diff₀ (ht : NullMeasurableSet t μ) (hfs : IntegrableOn f s μ) :
    ∫ x in s ∩ t, f x ∂μ + ∫ x in s \ t, f x ∂μ = ∫ x in s, f x ∂μ 
goal: ∫ (x : X) in s ∩ t, f x ∂μ + ∫ (x : X) in s \ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.integral_inter_add_diff
code: theorem integral_inter_add_diff (ht : MeasurableSet t) (hfs : IntegrableOn f s μ) :
    ∫ x in s ∩ t, f x ∂μ + ∫ x in s \ t, f x ∂μ = ∫ x in s, f x ∂μ 
goal: ∫ (x : X) in s ∩ t, f x ∂μ + ∫ (x : X) in s \ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.integral_finset_biUnion
code: theorem integral_finset_biUnion {ι : Type*} (t : Finset ι) {s : ι → Set X}
    (hs : ∀ i ∈ t, MeasurableSet (s i)) (h's : Set.Pairwise (↑t) (Disjoint on s))
    (hf : ∀ i ∈ t, IntegrableOn f (s i) μ) :
    ∫ x in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ x in s i, f x ∂μ 
goal: ∫ (x : X) in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ (x : X) in s i, f x ∂μ

full_name: MeasureTheory.integral_fintype_iUnion
code: theorem integral_fintype_iUnion {ι : Type*} [Fintype ι] {s : ι → Set X}
    (hs : ∀ i, MeasurableSet (s i)) (h's : Pairwise (Disjoint on s))
    (hf : ∀ i, IntegrableOn f (s i) μ) : ∫ x in ⋃ i, s i, f x ∂μ = ∑ i, ∫ x in s i, f x ∂μ 
goal: ∫ (x : X) in ⋃ i, s i, f x ∂μ = ∑ i : ι, ∫ (x : X) in s i, f x ∂μ

full_name: MeasureTheory.integral_empty
code: theorem integral_empty : ∫ x in ∅, f x ∂μ = 0 
goal: ∫ (x : X) in ∅, f x ∂μ = 0

full_name: MeasureTheory.integral_univ
code: theorem integral_univ : ∫ x in univ, f x ∂μ = ∫ x, f x ∂μ 
goal: ∫ (x : X) in Set.univ, f x ∂μ = ∫ (x : X), f x ∂μ

full_name: MeasureTheory.integral_add_compl₀
code: theorem integral_add_compl₀ (hs : NullMeasurableSet s μ) (hfi : Integrable f μ) :
    ∫ x in s, f x ∂μ + ∫ x in sᶜ, f x ∂μ = ∫ x, f x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, f x ∂μ = ∫ (x : X), f x ∂μ

full_name: MeasureTheory.integral_add_compl
code: theorem integral_add_compl (hs : MeasurableSet s) (hfi : Integrable f μ) :
    ∫ x in s, f x ∂μ + ∫ x in sᶜ, f x ∂μ = ∫ x, f x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, f x ∂μ = ∫ (x : X), f x ∂μ

full_name: MeasureTheory.setIntegral_compl
code: theorem setIntegral_compl (hs : MeasurableSet s) (hfi : Integrable f μ) :
    ∫ x in sᶜ, f x ∂μ = ∫ x, f x ∂μ - ∫ x in s, f x ∂μ 
goal: ∫ (x : X) in sᶜ, f x ∂μ = ∫ (x : X), f x ∂μ - ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.integral_indicator
code: /-- For a function `f` and a measurable set `s`, the integral of `indicator s f`
over the whole space is equal to `∫ x in s, f x ∂μ` defined as `∫ x, f x ∂(μ.restrict s)`. -/
theorem integral_indicator (hs : MeasurableSet s) :
    ∫ x, indicator s f x ∂μ = ∫ x in s, f x ∂μ 
goal: ∫ (x : X), s.indicator f x ∂μ = ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.setIntegral_indicator
code: theorem setIntegral_indicator (ht : MeasurableSet t) :
    ∫ x in s, t.indicator f x ∂μ = ∫ x in s ∩ t, f x ∂μ 
goal: ∫ (x : X) in s, t.indicator f x ∂μ = ∫ (x : X) in s ∩ t, f x ∂μ

full_name: MeasureTheory.ofReal_setIntegral_one_of_measure_ne_top
code: theorem ofReal_setIntegral_one_of_measure_ne_top {X : Type*} {m : MeasurableSpace X}
    {μ : Measure X} {s : Set X} (hs : μ s ≠ ∞) : ENNReal.ofReal (∫ _ in s, (1 : ℝ) ∂μ) = μ s 
goal: ENNReal.ofReal (∫ (x : X) in s, 1 ∂μ) = μ s

full_name: MeasureTheory.ofReal_setIntegral_one
code: theorem ofReal_setIntegral_one {X : Type*} {_ : MeasurableSpace X} (μ : Measure X)
    [IsFiniteMeasure μ] (s : Set X) : ENNReal.ofReal (∫ _ in s, (1 : ℝ) ∂μ) = μ s 
goal: ∀ {x : MeasurableSpace X} (μ : MeasureTheory.Measure X) [inst : MeasureTheory.IsFiniteMeasure μ] (s : Set X),
  ENNReal.ofReal (∫ (x : X) in s, 1 ∂μ) = μ s

full_name: MeasureTheory.integral_piecewise
code: theorem integral_piecewise [DecidablePred (· ∈ s)] (hs : MeasurableSet s) (hf : IntegrableOn f s μ)
    (hg : IntegrableOn g sᶜ μ) :
    ∫ x, s.piecewise f g x ∂μ = ∫ x in s, f x ∂μ + ∫ x in sᶜ, g x ∂μ 
goal: ∫ (x : X), s.piecewise f g x ∂μ = ∫ (x : X) in s, f x ∂μ + ∫ (x : X) in sᶜ, g x ∂μ

full_name: MeasureTheory.tendsto_setIntegral_of_monotone
code: theorem tendsto_setIntegral_of_monotone {ι : Type*} [Countable ι] [SemilatticeSup ι]
    {s : ι → Set X} (hsm : ∀ i, MeasurableSet (s i)) (h_mono : Monotone s)
    (hfi : IntegrableOn f (⋃ n, s n) μ) :
    Tendsto (fun i => ∫ x in s i, f x ∂μ) atTop (𝓝 (∫ x in ⋃ n, s n, f x ∂μ)) 
goal: Filter.Tendsto (fun i => ∫ (x : X) in s i, f x ∂μ) Filter.atTop (𝓝 (∫ (x : X) in ⋃ n, s n, f x ∂μ))

full_name: MeasureTheory.tendsto_setIntegral_of_antitone
code: theorem tendsto_setIntegral_of_antitone {ι : Type*} [Countable ι] [SemilatticeSup ι]
    {s : ι → Set X} (hsm : ∀ i, MeasurableSet (s i)) (h_anti : Antitone s)
    (hfi : ∃ i, IntegrableOn f (s i) μ) :
    Tendsto (fun i ↦ ∫ x in s i, f x ∂μ) atTop (𝓝 (∫ x in ⋂ n, s n, f x ∂μ)) 
goal: Filter.Tendsto (fun i => ∫ (x : X) in s i, f x ∂μ) Filter.atTop (𝓝 (∫ (x : X) in ⋂ n, s n, f x ∂μ))

full_name: MeasureTheory.hasSum_integral_iUnion_ae
code: theorem hasSum_integral_iUnion_ae {ι : Type*} [Countable ι] {s : ι → Set X}
    (hm : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AEDisjoint μ on s))
    (hfi : IntegrableOn f (⋃ i, s i) μ) :
    HasSum (fun n => ∫ x in s n, f x ∂μ) (∫ x in ⋃ n, s n, f x ∂μ) 
goal: HasSum (fun n => ∫ (x : X) in s n, f x ∂μ) (∫ (x : X) in ⋃ n, s n, f x ∂μ)

full_name: MeasureTheory.hasSum_integral_iUnion
code: theorem hasSum_integral_iUnion {ι : Type*} [Countable ι] {s : ι → Set X}
    (hm : ∀ i, MeasurableSet (s i)) (hd : Pairwise (Disjoint on s))
    (hfi : IntegrableOn f (⋃ i, s i) μ) :
    HasSum (fun n => ∫ x in s n, f x ∂μ) (∫ x in ⋃ n, s n, f x ∂μ) 
goal: HasSum (fun n => ∫ (x : X) in s n, f x ∂μ) (∫ (x : X) in ⋃ n, s n, f x ∂μ)

full_name: MeasureTheory.integral_iUnion
code: theorem integral_iUnion {ι : Type*} [Countable ι] {s : ι → Set X} (hm : ∀ i, MeasurableSet (s i))
    (hd : Pairwise (Disjoint on s)) (hfi : IntegrableOn f (⋃ i, s i) μ) :
    ∫ x in ⋃ n, s n, f x ∂μ = ∑' n, ∫ x in s n, f x ∂μ 
goal: ∫ (x : X) in ⋃ n, s n, f x ∂μ = ∑' (n : ι), ∫ (x : X) in s n, f x ∂μ

full_name: MeasureTheory.integral_iUnion_ae
code: theorem integral_iUnion_ae {ι : Type*} [Countable ι] {s : ι → Set X}
    (hm : ∀ i, NullMeasurableSet (s i) μ) (hd : Pairwise (AEDisjoint μ on s))
    (hfi : IntegrableOn f (⋃ i, s i) μ) : ∫ x in ⋃ n, s n, f x ∂μ = ∑' n, ∫ x in s n, f x ∂μ 
goal: ∫ (x : X) in ⋃ n, s n, f x ∂μ = ∑' (n : ι), ∫ (x : X) in s n, f x ∂μ

full_name: MeasureTheory.setIntegral_eq_zero_of_ae_eq_zero
code: theorem setIntegral_eq_zero_of_ae_eq_zero (ht_eq : ∀ᵐ x ∂μ, x ∈ t → f x = 0) :
    ∫ x in t, f x ∂μ = 0 
goal: ∫ (x : X) in t, f x ∂μ = 0

full_name: MeasureTheory.setIntegral_eq_zero_of_forall_eq_zero
code: theorem setIntegral_eq_zero_of_forall_eq_zero (ht_eq : ∀ x ∈ t, f x = 0) :
    ∫ x in t, f x ∂μ = 0 
goal: ∫ (x : X) in t, f x ∂μ = 0

full_name: MeasureTheory.integral_union_eq_left_of_ae_aux
code: theorem integral_union_eq_left_of_ae_aux (ht_eq : ∀ᵐ x ∂μ.restrict t, f x = 0)
    (haux : StronglyMeasurable f) (H : IntegrableOn f (s ∪ t) μ) :
    ∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ 
goal: ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.integral_union_eq_left_of_ae
code: theorem integral_union_eq_left_of_ae (ht_eq : ∀ᵐ x ∂μ.restrict t, f x = 0) :
    ∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ 
goal: ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.integral_union_eq_left_of_forall₀
code: theorem integral_union_eq_left_of_forall₀ {f : X → E} (ht : NullMeasurableSet t μ)
    (ht_eq : ∀ x ∈ t, f x = 0) : ∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ 
goal: ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.integral_union_eq_left_of_forall
code: theorem integral_union_eq_left_of_forall {f : X → E} (ht : MeasurableSet t)
    (ht_eq : ∀ x ∈ t, f x = 0) : ∫ x in s ∪ t, f x ∂μ = ∫ x in s, f x ∂μ 
goal: ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.setIntegral_eq_of_subset_of_ae_diff_eq_zero_aux
code: theorem setIntegral_eq_of_subset_of_ae_diff_eq_zero_aux (hts : s ⊆ t)
    (h't : ∀ᵐ x ∂μ, x ∈ t \ s → f x = 0) (haux : StronglyMeasurable f)
    (h'aux : IntegrableOn f t μ) : ∫ x in t, f x ∂μ = ∫ x in s, f x ∂μ 
goal: ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.setIntegral_eq_of_subset_of_ae_diff_eq_zero
code: /-- If a function vanishes almost everywhere on `t \ s` with `s ⊆ t`, then its integrals on `s`
and `t` coincide if `t` is null-measurable. -/
theorem setIntegral_eq_of_subset_of_ae_diff_eq_zero (ht : NullMeasurableSet t μ) (hts : s ⊆ t)
    (h't : ∀ᵐ x ∂μ, x ∈ t \ s → f x = 0) : ∫ x in t, f x ∂μ = ∫ x in s, f x ∂μ 
goal: ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.setIntegral_eq_of_subset_of_forall_diff_eq_zero
code: /-- If a function vanishes on `t \ s` with `s ⊆ t`, then its integrals on `s`
and `t` coincide if `t` is measurable. -/
theorem setIntegral_eq_of_subset_of_forall_diff_eq_zero (ht : MeasurableSet t) (hts : s ⊆ t)
    (h't : ∀ x ∈ t \ s, f x = 0) : ∫ x in t, f x ∂μ = ∫ x in s, f x ∂μ 
goal: ∫ (x : X) in t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.setIntegral_eq_integral_of_ae_compl_eq_zero
code: /-- If a function vanishes almost everywhere on `sᶜ`, then its integral on `s`
coincides with its integral on the whole space. -/
theorem setIntegral_eq_integral_of_ae_compl_eq_zero (h : ∀ᵐ x ∂μ, x ∉ s → f x = 0) :
    ∫ x in s, f x ∂μ = ∫ x, f x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X), f x ∂μ

full_name: MeasureTheory.setIntegral_eq_integral_of_forall_compl_eq_zero
code: /-- If a function vanishes on `sᶜ`, then its integral on `s` coincides with its integral on the
whole space. -/
theorem setIntegral_eq_integral_of_forall_compl_eq_zero (h : ∀ x, x ∉ s → f x = 0) :
    ∫ x in s, f x ∂μ = ∫ x, f x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X), f x ∂μ

full_name: MeasureTheory.setIntegral_neg_eq_setIntegral_nonpos
code: theorem setIntegral_neg_eq_setIntegral_nonpos [LinearOrder E] {f : X → E}
    (hf : AEStronglyMeasurable f μ) :
    ∫ x in {x | f x < 0}, f x ∂μ = ∫ x in {x | f x ≤ 0}, f x ∂μ 
goal: ∫ (x : X) in {x | f x < 0}, f x ∂μ = ∫ (x : X) in {x | f x ≤ 0}, f x ∂μ

full_name: MeasureTheory.integral_norm_eq_pos_sub_neg
code: theorem integral_norm_eq_pos_sub_neg {f : X → ℝ} (hfi : Integrable f μ) :
    ∫ x, ‖f x‖ ∂μ = ∫ x in {x | 0 ≤ f x}, f x ∂μ - ∫ x in {x | f x ≤ 0}, f x ∂μ 
goal: ∫ (x : X), ‖f x‖ ∂μ = ∫ (x : X) in {x | 0 ≤ f x}, f x ∂μ - ∫ (x : X) in {x | f x ≤ 0}, f x ∂μ

full_name: MeasureTheory.setIntegral_const
code: theorem setIntegral_const [CompleteSpace E] (c : E) : ∫ _ in s, c ∂μ = (μ s).toReal • c 
goal: ∫ (x : X) in s, c ∂μ = (μ s).toReal • c

full_name: MeasureTheory.integral_indicator_const
code: @[simp]
theorem integral_indicator_const [CompleteSpace E] (e : E) ⦃s : Set X⦄ (s_meas : MeasurableSet s) :
    ∫ x : X, s.indicator (fun _ : X => e) x ∂μ = (μ s).toReal • e 
goal: ∫ (x : X), s.indicator (fun x => e) x ∂μ = (μ s).toReal • e

full_name: MeasureTheory.integral_indicator_one
code: @[simp]
theorem integral_indicator_one ⦃s : Set X⦄ (hs : MeasurableSet s) :
    ∫ x, s.indicator 1 x ∂μ = (μ s).toReal 
goal: ∫ (x : X), s.indicator 1 x ∂μ = (μ s).toReal

full_name: MeasureTheory.setIntegral_indicatorConstLp
code: theorem setIntegral_indicatorConstLp [CompleteSpace E]
    {p : ℝ≥0∞} (hs : MeasurableSet s) (ht : MeasurableSet t) (hμt : μ t ≠ ∞) (e : E) :
    ∫ x in s, indicatorConstLp p ht hμt e x ∂μ = (μ (t ∩ s)).toReal • e 
goal: ∫ (x : X) in s, ↑↑(MeasureTheory.indicatorConstLp p ht hμt e) x ∂μ = (μ (t ∩ s)).toReal • e

full_name: MeasureTheory.integral_indicatorConstLp
code: theorem integral_indicatorConstLp [CompleteSpace E]
    {p : ℝ≥0∞} (ht : MeasurableSet t) (hμt : μ t ≠ ∞) (e : E) :
    ∫ x, indicatorConstLp p ht hμt e x ∂μ = (μ t).toReal • e 
goal: ∫ (x : X), ↑↑(MeasureTheory.indicatorConstLp p ht hμt e) x ∂μ = (μ t).toReal • e

full_name: MeasureTheory.setIntegral_map
code: theorem setIntegral_map {Y} [MeasurableSpace Y] {g : X → Y} {f : Y → E} {s : Set Y}
    (hs : MeasurableSet s) (hf : AEStronglyMeasurable f (Measure.map g μ)) (hg : AEMeasurable g μ) :
    ∫ y in s, f y ∂Measure.map g μ = ∫ x in g ⁻¹' s, f (g x) ∂μ 
goal: ∫ (y : Y) in s, f y ∂MeasureTheory.Measure.map g μ = ∫ (x : X) in g ⁻¹' s, f (g x) ∂μ

full_name: MeasurableEmbedding.setIntegral_map
code: theorem _root_.MeasurableEmbedding.setIntegral_map {Y} {_ : MeasurableSpace Y} {f : X → Y}
    (hf : MeasurableEmbedding f) (g : Y → E) (s : Set Y) :
    ∫ y in s, g y ∂Measure.map f μ = ∫ x in f ⁻¹' s, g (f x) ∂μ 
goal: ∀ {x : MeasurableSpace Y} {f : X → Y},
  MeasurableEmbedding f →
    ∀ (g : Y → E) (s : Set Y), ∫ (y : Y) in s, g y ∂MeasureTheory.Measure.map f μ = ∫ (x : X) in f ⁻¹' s, g (f x) ∂μ

full_name: ClosedEmbedding.setIntegral_map
code: theorem _root_.ClosedEmbedding.setIntegral_map [TopologicalSpace X] [BorelSpace X] {Y}
    [MeasurableSpace Y] [TopologicalSpace Y] [BorelSpace Y] {g : X → Y} {f : Y → E} (s : Set Y)
    (hg : ClosedEmbedding g) : ∫ y in s, f y ∂Measure.map g μ = ∫ x in g ⁻¹' s, f (g x) ∂μ 
goal: ∫ (y : Y) in s, f y ∂MeasureTheory.Measure.map g μ = ∫ (x : X) in g ⁻¹' s, f (g x) ∂μ

full_name: MeasureTheory.MeasurePreserving.setIntegral_preimage_emb
code: theorem MeasurePreserving.setIntegral_preimage_emb {Y} {_ : MeasurableSpace Y} {f : X → Y} {ν}
    (h₁ : MeasurePreserving f μ ν) (h₂ : MeasurableEmbedding f) (g : Y → E) (s : Set Y) :
    ∫ x in f ⁻¹' s, g (f x) ∂μ = ∫ y in s, g y ∂ν 
goal: ∀ {x : MeasurableSpace Y} {f : X → Y} {ν : MeasureTheory.Measure Y},
  MeasureTheory.MeasurePreserving f μ ν →
    MeasurableEmbedding f → ∀ (g : Y → E) (s : Set Y), ∫ (x : X) in f ⁻¹' s, g (f x) ∂μ = ∫ (y : Y) in s, g y ∂ν

full_name: MeasureTheory.MeasurePreserving.setIntegral_image_emb
code: theorem MeasurePreserving.setIntegral_image_emb {Y} {_ : MeasurableSpace Y} {f : X → Y} {ν}
    (h₁ : MeasurePreserving f μ ν) (h₂ : MeasurableEmbedding f) (g : Y → E) (s : Set X) :
    ∫ y in f '' s, g y ∂ν = ∫ x in s, g (f x) ∂μ 
goal: ∀ {x : MeasurableSpace Y} {f : X → Y} {ν : MeasureTheory.Measure Y},
  MeasureTheory.MeasurePreserving f μ ν →
    MeasurableEmbedding f → ∀ (g : Y → E) (s : Set X), ∫ (y : Y) in f '' s, g y ∂ν = ∫ (x : X) in s, g (f x) ∂μ

full_name: MeasureTheory.setIntegral_map_equiv
code: theorem setIntegral_map_equiv {Y} [MeasurableSpace Y] (e : X ≃ᵐ Y) (f : Y → E) (s : Set Y) :
    ∫ y in s, f y ∂Measure.map e μ = ∫ x in e ⁻¹' s, f (e x) ∂μ 
goal: ∫ (y : Y) in s, f y ∂MeasureTheory.Measure.map (⇑e) μ = ∫ (x : X) in ⇑e ⁻¹' s, f (e x) ∂μ

full_name: MeasureTheory.norm_setIntegral_le_of_norm_le_const_ae
code: theorem norm_setIntegral_le_of_norm_le_const_ae {C : ℝ} (hs : μ s < ∞)
    (hC : ∀ᵐ x ∂μ.restrict s, ‖f x‖ ≤ C) : ‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal 
goal: ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * (μ s).toReal

full_name: MeasureTheory.norm_setIntegral_le_of_norm_le_const_ae'
code: theorem norm_setIntegral_le_of_norm_le_const_ae' {C : ℝ} (hs : μ s < ∞)
    (hC : ∀ᵐ x ∂μ, x ∈ s → ‖f x‖ ≤ C) (hfm : AEStronglyMeasurable f (μ.restrict s)) :
    ‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal 
goal: ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * (μ s).toReal

full_name: MeasureTheory.norm_setIntegral_le_of_norm_le_const_ae''
code: theorem norm_setIntegral_le_of_norm_le_const_ae'' {C : ℝ} (hs : μ s < ∞) (hsm : MeasurableSet s)
    (hC : ∀ᵐ x ∂μ, x ∈ s → ‖f x‖ ≤ C) : ‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal 
goal: ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * (μ s).toReal

full_name: MeasureTheory.norm_setIntegral_le_of_norm_le_const
code: theorem norm_setIntegral_le_of_norm_le_const {C : ℝ} (hs : μ s < ∞) (hC : ∀ x ∈ s, ‖f x‖ ≤ C)
    (hfm : AEStronglyMeasurable f (μ.restrict s)) : ‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal 
goal: ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * (μ s).toReal

full_name: MeasureTheory.norm_setIntegral_le_of_norm_le_const'
code: theorem norm_setIntegral_le_of_norm_le_const' {C : ℝ} (hs : μ s < ∞) (hsm : MeasurableSet s)
    (hC : ∀ x ∈ s, ‖f x‖ ≤ C) : ‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal 
goal: ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * (μ s).toReal

full_name: MeasureTheory.setIntegral_eq_zero_iff_of_nonneg_ae
code: theorem setIntegral_eq_zero_iff_of_nonneg_ae {f : X → ℝ} (hf : 0 ≤ᵐ[μ.restrict s] f)
    (hfi : IntegrableOn f s μ) : ∫ x in s, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict s] 0 
goal: ∫ (x : X) in s, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict s] 0

full_name: MeasureTheory.setIntegral_pos_iff_support_of_nonneg_ae
code: theorem setIntegral_pos_iff_support_of_nonneg_ae {f : X → ℝ} (hf : 0 ≤ᵐ[μ.restrict s] f)
    (hfi : IntegrableOn f s μ) : (0 < ∫ x in s, f x ∂μ) ↔ 0 < μ (support f ∩ s) 
goal: 0 < ∫ (x : X) in s, f x ∂μ ↔ 0 < μ (Function.support f ∩ s)

full_name: MeasureTheory.setIntegral_gt_gt
code: theorem setIntegral_gt_gt {R : ℝ} {f : X → ℝ} (hR : 0 ≤ R)
    (hfint : IntegrableOn f {x | ↑R < f x} μ) (hμ : μ {x | ↑R < f x} ≠ 0) :
    (μ {x | ↑R < f x}).toReal * R < ∫ x in {x | ↑R < f x}, f x ∂μ 
goal: (μ {x | R < f x}).toReal * R < ∫ (x : X) in {x | R < f x}, f x ∂μ

full_name: MeasureTheory.setIntegral_trim
code: theorem setIntegral_trim {X} {m m0 : MeasurableSpace X} {μ : Measure X} (hm : m ≤ m0) {f : X → E}
    (hf_meas : StronglyMeasurable[m] f) {s : Set X} (hs : MeasurableSet[m] s) :
    ∫ x in s, f x ∂μ = ∫ x in s, f x ∂μ.trim hm 
goal: ∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, f x ∂μ.trim hm

full_name: MeasureTheory.integral_Icc_eq_integral_Ioc'
code: theorem integral_Icc_eq_integral_Ioc' (hx : μ {x} = 0) :
    ∫ t in Icc x y, f t ∂μ = ∫ t in Ioc x y, f t ∂μ 
goal: ∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ioc x y, f t ∂μ

full_name: MeasureTheory.integral_Icc_eq_integral_Ico'
code: theorem integral_Icc_eq_integral_Ico' (hy : μ {y} = 0) :
    ∫ t in Icc x y, f t ∂μ = ∫ t in Ico x y, f t ∂μ 
goal: ∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ico x y, f t ∂μ

full_name: MeasureTheory.integral_Ioc_eq_integral_Ioo'
code: theorem integral_Ioc_eq_integral_Ioo' (hy : μ {y} = 0) :
    ∫ t in Ioc x y, f t ∂μ = ∫ t in Ioo x y, f t ∂μ 
goal: ∫ (t : X) in Set.Ioc x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ

full_name: MeasureTheory.integral_Ico_eq_integral_Ioo'
code: theorem integral_Ico_eq_integral_Ioo' (hx : μ {x} = 0) :
    ∫ t in Ico x y, f t ∂μ = ∫ t in Ioo x y, f t ∂μ 
goal: ∫ (t : X) in Set.Ico x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ

full_name: MeasureTheory.integral_Icc_eq_integral_Ioo'
code: theorem integral_Icc_eq_integral_Ioo' (hx : μ {x} = 0) (hy : μ {y} = 0) :
    ∫ t in Icc x y, f t ∂μ = ∫ t in Ioo x y, f t ∂μ 
goal: ∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ

full_name: MeasureTheory.integral_Iic_eq_integral_Iio'
code: theorem integral_Iic_eq_integral_Iio' (hx : μ {x} = 0) :
    ∫ t in Iic x, f t ∂μ = ∫ t in Iio x, f t ∂μ 
goal: ∫ (t : X) in Set.Iic x, f t ∂μ = ∫ (t : X) in Set.Iio x, f t ∂μ

full_name: MeasureTheory.integral_Ici_eq_integral_Ioi'
code: theorem integral_Ici_eq_integral_Ioi' (hx : μ {x} = 0) :
    ∫ t in Ici x, f t ∂μ = ∫ t in Ioi x, f t ∂μ 
goal: ∫ (t : X) in Set.Ici x, f t ∂μ = ∫ (t : X) in Set.Ioi x, f t ∂μ

full_name: MeasureTheory.integral_Icc_eq_integral_Ioc
code: theorem integral_Icc_eq_integral_Ioc : ∫ t in Icc x y, f t ∂μ = ∫ t in Ioc x y, f t ∂μ 
goal: ∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ioc x y, f t ∂μ

full_name: MeasureTheory.integral_Icc_eq_integral_Ico
code: theorem integral_Icc_eq_integral_Ico : ∫ t in Icc x y, f t ∂μ = ∫ t in Ico x y, f t ∂μ 
goal: ∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ico x y, f t ∂μ

full_name: MeasureTheory.integral_Ioc_eq_integral_Ioo
code: theorem integral_Ioc_eq_integral_Ioo : ∫ t in Ioc x y, f t ∂μ = ∫ t in Ioo x y, f t ∂μ 
goal: ∫ (t : X) in Set.Ioc x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ

full_name: MeasureTheory.integral_Ico_eq_integral_Ioo
code: theorem integral_Ico_eq_integral_Ioo : ∫ t in Ico x y, f t ∂μ = ∫ t in Ioo x y, f t ∂μ 
goal: ∫ (t : X) in Set.Ico x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ

full_name: MeasureTheory.integral_Icc_eq_integral_Ioo
code: theorem integral_Icc_eq_integral_Ioo : ∫ t in Icc x y, f t ∂μ = ∫ t in Ioo x y, f t ∂μ 
goal: ∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ioo x y, f t ∂μ

full_name: MeasureTheory.integral_Iic_eq_integral_Iio
code: theorem integral_Iic_eq_integral_Iio : ∫ t in Iic x, f t ∂μ = ∫ t in Iio x, f t ∂μ 
goal: ∫ (t : X) in Set.Iic x, f t ∂μ = ∫ (t : X) in Set.Iio x, f t ∂μ

full_name: MeasureTheory.integral_Ici_eq_integral_Ioi
code: theorem integral_Ici_eq_integral_Ioi : ∫ t in Ici x, f t ∂μ = ∫ t in Ioi x, f t ∂μ 
goal: ∫ (t : X) in Set.Ici x, f t ∂μ = ∫ (t : X) in Set.Ioi x, f t ∂μ

full_name: MeasureTheory.setIntegral_mono_ae_restrict
code: theorem setIntegral_mono_ae_restrict (h : f ≤ᵐ[μ.restrict s] g) :
    ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

full_name: MeasureTheory.setIntegral_mono_ae
code: theorem setIntegral_mono_ae (h : f ≤ᵐ[μ] g) : ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

full_name: MeasureTheory.setIntegral_mono_on
code: theorem setIntegral_mono_on (hs : MeasurableSet s) (h : ∀ x ∈ s, f x ≤ g x) :
    ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

full_name: MeasureTheory.setIntegral_mono_on_ae
code: theorem setIntegral_mono_on_ae (hs : MeasurableSet s) (h : ∀ᵐ x ∂μ, x ∈ s → f x ≤ g x) :
    ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

full_name: MeasureTheory.setIntegral_mono
code: theorem setIntegral_mono (h : f ≤ g) : ∫ x in s, f x ∂μ ≤ ∫ x in s, g x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in s, g x ∂μ

full_name: MeasureTheory.setIntegral_mono_set
code: theorem setIntegral_mono_set (hfi : IntegrableOn f t μ) (hf : 0 ≤ᵐ[μ.restrict t] f)
    (hst : s ≤ᵐ[μ] t) : ∫ x in s, f x ∂μ ≤ ∫ x in t, f x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in t, f x ∂μ

full_name: MeasureTheory.setIntegral_le_integral
code: theorem setIntegral_le_integral (hfi : Integrable f μ) (hf : 0 ≤ᵐ[μ] f) :
    ∫ x in s, f x ∂μ ≤ ∫ x, f x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X), f x ∂μ

full_name: MeasureTheory.setIntegral_ge_of_const_le
code: theorem setIntegral_ge_of_const_le {c : ℝ} (hs : MeasurableSet s) (hμs : μ s ≠ ∞)
    (hf : ∀ x ∈ s, c ≤ f x) (hfint : IntegrableOn (fun x : X => f x) s μ) :
    c * (μ s).toReal ≤ ∫ x in s, f x ∂μ 
goal: c * (μ s).toReal ≤ ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.setIntegral_nonneg_of_ae_restrict
code: theorem setIntegral_nonneg_of_ae_restrict (hf : 0 ≤ᵐ[μ.restrict s] f) : 0 ≤ ∫ x in s, f x ∂μ 
goal: 0 ≤ ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.setIntegral_nonneg_of_ae
code: theorem setIntegral_nonneg_of_ae (hf : 0 ≤ᵐ[μ] f) : 0 ≤ ∫ x in s, f x ∂μ 
goal: 0 ≤ ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.setIntegral_nonneg
code: theorem setIntegral_nonneg (hs : MeasurableSet s) (hf : ∀ x, x ∈ s → 0 ≤ f x) :
    0 ≤ ∫ x in s, f x ∂μ 
goal: 0 ≤ ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.setIntegral_nonneg_ae
code: theorem setIntegral_nonneg_ae (hs : MeasurableSet s) (hf : ∀ᵐ x ∂μ, x ∈ s → 0 ≤ f x) :
    0 ≤ ∫ x in s, f x ∂μ 
goal: 0 ≤ ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.setIntegral_le_nonneg
code: theorem setIntegral_le_nonneg {s : Set X} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
    (hfi : Integrable f μ) : ∫ x in s, f x ∂μ ≤ ∫ x in {y | 0 ≤ f y}, f x ∂μ 
goal: ∫ (x : X) in s, f x ∂μ ≤ ∫ (x : X) in {y | 0 ≤ f y}, f x ∂μ

full_name: MeasureTheory.setIntegral_nonpos_of_ae_restrict
code: theorem setIntegral_nonpos_of_ae_restrict (hf : f ≤ᵐ[μ.restrict s] 0) : ∫ x in s, f x ∂μ ≤ 0 
goal: ∫ (x : X) in s, f x ∂μ ≤ 0

full_name: MeasureTheory.setIntegral_nonpos_of_ae
code: theorem setIntegral_nonpos_of_ae (hf : f ≤ᵐ[μ] 0) : ∫ x in s, f x ∂μ ≤ 0 
goal: ∫ (x : X) in s, f x ∂μ ≤ 0

full_name: MeasureTheory.setIntegral_nonpos_ae
code: theorem setIntegral_nonpos_ae (hs : MeasurableSet s) (hf : ∀ᵐ x ∂μ, x ∈ s → f x ≤ 0) :
    ∫ x in s, f x ∂μ ≤ 0 
goal: ∫ (x : X) in s, f x ∂μ ≤ 0

full_name: MeasureTheory.setIntegral_nonpos
code: theorem setIntegral_nonpos (hs : MeasurableSet s) (hf : ∀ x, x ∈ s → f x ≤ 0) :
    ∫ x in s, f x ∂μ ≤ 0 
goal: ∫ (x : X) in s, f x ∂μ ≤ 0

full_name: MeasureTheory.setIntegral_nonpos_le
code: theorem setIntegral_nonpos_le {s : Set X} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
    (hfi : Integrable f μ) : ∫ x in {y | f y ≤ 0}, f x ∂μ ≤ ∫ x in s, f x ∂μ 
goal: ∫ (x : X) in {y | f y ≤ 0}, f x ∂μ ≤ ∫ (x : X) in s, f x ∂μ

full_name: MeasureTheory.Integrable.measure_le_integral
code: lemma Integrable.measure_le_integral {f : X → ℝ} (f_int : Integrable f μ) (f_nonneg : 0 ≤ᵐ[μ] f)
    {s : Set X} (hs : ∀ x ∈ s, 1 ≤ f x) :
    μ s ≤ ENNReal.ofReal (∫ x, f x ∂μ) 
goal: μ s ≤ ENNReal.ofReal (∫ (x : X), f x ∂μ)

full_name: MeasureTheory.integral_le_measure
code: lemma integral_le_measure {f : X → ℝ} {s : Set X}
    (hs : ∀ x ∈ s, f x ≤ 1) (h's : ∀ x ∈ sᶜ, f x ≤ 0) :
    ENNReal.ofReal (∫ x, f x ∂μ) ≤ μ s 
goal: ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ μ s

full_name: MeasureTheory.integrableOn_iUnion_of_summable_integral_norm
code: theorem integrableOn_iUnion_of_summable_integral_norm {f : X → E} {s : ι → Set X}
    (hs : ∀ i : ι, MeasurableSet (s i)) (hi : ∀ i : ι, IntegrableOn f (s i) μ)
    (h : Summable fun i : ι => ∫ x : X in s i, ‖f x‖ ∂μ) : IntegrableOn f (iUnion s) μ 
goal: MeasureTheory.IntegrableOn f (Set.iUnion s) μ

full_name: MeasureTheory.integrableOn_iUnion_of_summable_norm_restrict
code: /-- If `s` is a countable family of compact sets, `f` is a continuous function, and the sequence
`‖f.restrict (s i)‖ * μ (s i)` is summable, then `f` is integrable on the union of the `s i`. -/
theorem integrableOn_iUnion_of_summable_norm_restrict {f : C(X, E)} {s : ι → Compacts X}
    (hf : Summable fun i : ι => ‖f.restrict (s i)‖ * ENNReal.toReal (μ <| s i)) :
    IntegrableOn f (⋃ i : ι, s i) μ 
goal: MeasureTheory.IntegrableOn (⇑f) (⋃ i, ↑(s i)) μ

full_name: MeasureTheory.integrable_of_summable_norm_restrict
code: /-- If `s` is a countable family of compact sets covering `X`, `f` is a continuous function, and
the sequence `‖f.restrict (s i)‖ * μ (s i)` is summable, then `f` is integrable. -/
theorem integrable_of_summable_norm_restrict {f : C(X, E)} {s : ι → Compacts X}
    (hf : Summable fun i : ι => ‖f.restrict (s i)‖ * ENNReal.toReal (μ <| s i))
    (hs : ⋃ i : ι, ↑(s i) = (univ : Set X)) : Integrable f μ 
goal: MeasureTheory.Integrable (⇑f) μ

full_name: MeasureTheory.Lp_toLp_restrict_add
code: /-- For `f : Lp E p μ`, we can define an element of `Lp E p (μ.restrict s)` by
`(Lp.memℒp f).restrict s).toLp f`. This map is additive. -/
theorem Lp_toLp_restrict_add (f g : Lp E p μ) (s : Set X) :
    ((Lp.memℒp (f + g)).restrict s).toLp (⇑(f + g)) =
      ((Lp.memℒp f).restrict s).toLp f + ((Lp.memℒp g).restrict s).toLp g 
goal: MeasureTheory.Memℒp.toLp ↑↑(f + g) ⋯ = MeasureTheory.Memℒp.toLp ↑↑f ⋯ + MeasureTheory.Memℒp.toLp ↑↑g ⋯

full_name: MeasureTheory.Lp_toLp_restrict_smul
code: /-- For `f : Lp E p μ`, we can define an element of `Lp E p (μ.restrict s)` by
`(Lp.memℒp f).restrict s).toLp f`. This map commutes with scalar multiplication. -/
theorem Lp_toLp_restrict_smul (c : 𝕜) (f : Lp F p μ) (s : Set X) :
    ((Lp.memℒp (c • f)).restrict s).toLp (⇑(c • f)) = c • ((Lp.memℒp f).restrict s).toLp f 
goal: MeasureTheory.Memℒp.toLp ↑↑(c • f) ⋯ = c • MeasureTheory.Memℒp.toLp ↑↑f ⋯

full_name: MeasureTheory.norm_Lp_toLp_restrict_le
code: /-- For `f : Lp E p μ`, we can define an element of `Lp E p (μ.restrict s)` by
`(Lp.memℒp f).restrict s).toLp f`. This map is non-expansive. -/
theorem norm_Lp_toLp_restrict_le (s : Set X) (f : Lp E p μ) :
    ‖((Lp.memℒp f).restrict s).toLp f‖ ≤ ‖f‖ 
goal: ‖MeasureTheory.Memℒp.toLp ↑↑f ⋯‖ ≤ ‖f‖

full_name: MeasureTheory.LpToLpRestrictCLM
code: /-- Continuous linear map sending a function of `Lp F p μ` to the same function in
`Lp F p (μ.restrict s)`. -/
def LpToLpRestrictCLM (μ : Measure X) (p : ℝ≥0∞) [hp : Fact (1 ≤ p)] (s : Set X) :
    Lp F p μ →L[𝕜] Lp F p (μ.restrict s) 
goal: ↥(MeasureTheory.Lp F p μ) →L[𝕜] ↥(MeasureTheory.Lp F p (μ.restrict s))

full_name: MeasureTheory.LpToLpRestrictCLM_coeFn
code: theorem LpToLpRestrictCLM_coeFn [Fact (1 ≤ p)] (s : Set X) (f : Lp F p μ) :
    LpToLpRestrictCLM X F 𝕜 μ p s f =ᵐ[μ.restrict s] f 
goal: ↑↑((MeasureTheory.LpToLpRestrictCLM X F 𝕜 μ p s) f) =ᵐ[μ.restrict s] ↑↑f

full_name: MeasureTheory.continuous_setIntegral
code: @[continuity]
theorem continuous_setIntegral [NormedSpace ℝ E] (s : Set X) :
    Continuous fun f : X →₁[μ] E => ∫ x in s, f x ∂μ 
goal: Continuous fun f => ∫ (x : X) in s, ↑↑f x ∂μ

full_name: Continuous.integral_pos_of_hasCompactSupport_nonneg_nonzero
code: theorem Continuous.integral_pos_of_hasCompactSupport_nonneg_nonzero [IsFiniteMeasureOnCompacts μ]
    {f : X → ℝ} {x : X} (f_cont : Continuous f) (f_comp : HasCompactSupport f) (f_nonneg : 0 ≤ f)
    (f_x : f x ≠ 0) : 0 < ∫ x, f x ∂μ 
goal: 0 < ∫ (x : X), f x ∂μ

full_name: Filter.Tendsto.integral_sub_linear_isLittleO_ae
code: /-- Fundamental theorem of calculus for set integrals:
if `μ` is a measure that is finite at a filter `l` and
`f` is a measurable function that has a finite limit `b` at `l ⊓ ae μ`, then
`∫ x in s i, f x ∂μ = μ (s i) • b + o(μ (s i))` at a filter `li` provided that
`s i` tends to `l.smallSets` along `li`.
Since `μ (s i)` is an `ℝ≥0∞` number, we use `(μ (s i)).toReal` in the actual statement.

Often there is a good formula for `(μ (s i)).toReal`, so the formalization can take an optional
argument `m` with this formula and a proof of `(fun i => (μ (s i)).toReal) =ᶠ[li] m`. Without these
arguments, `m i = (μ (s i)).toReal` is used in the output. -/
theorem Filter.Tendsto.integral_sub_linear_isLittleO_ae
    {μ : Measure X} {l : Filter X} [l.IsMeasurablyGenerated] {f : X → E} {b : E}
    (h : Tendsto f (l ⊓ ae μ) (𝓝 b)) (hfm : StronglyMeasurableAtFilter f l μ)
    (hμ : μ.FiniteAtFilter l) {s : ι → Set X} {li : Filter ι} (hs : Tendsto s li l.smallSets)
    (m : ι → ℝ := fun i => (μ (s i)).toReal)
    (hsμ : (fun i => (μ (s i)).toReal) =ᶠ[li] m := by rfl) :
    (fun i => (∫ x in s i, f x ∂μ) - m i • b) =o[li] m 
goal: (fun i => ∫ (x : X) in s i, f x ∂μ - m i • b) =o[li] m

full_name: ContinuousWithinAt.integral_sub_linear_isLittleO_ae
code: /-- Fundamental theorem of calculus for set integrals, `nhdsWithin` version: if `μ` is a locally
finite measure and `f` is an almost everywhere measurable function that is continuous at a point `a`
within a measurable set `t`, then `∫ x in s i, f x ∂μ = μ (s i) • f a + o(μ (s i))` at a filter `li`
provided that `s i` tends to `(𝓝[t] a).smallSets` along `li`.  Since `μ (s i)` is an `ℝ≥0∞`
number, we use `(μ (s i)).toReal` in the actual statement.

Often there is a good formula for `(μ (s i)).toReal`, so the formalization can take an optional
argument `m` with this formula and a proof of `(fun i => (μ (s i)).toReal) =ᶠ[li] m`. Without these
arguments, `m i = (μ (s i)).toReal` is used in the output. -/
theorem ContinuousWithinAt.integral_sub_linear_isLittleO_ae [TopologicalSpace X]
    [OpensMeasurableSpace X] {μ : Measure X}
    [IsLocallyFiniteMeasure μ] {x : X} {t : Set X} {f : X → E} (hx : ContinuousWithinAt f t x)
    (ht : MeasurableSet t) (hfm : StronglyMeasurableAtFilter f (𝓝[t] x) μ) {s : ι → Set X}
    {li : Filter ι} (hs : Tendsto s li (𝓝[t] x).smallSets) (m : ι → ℝ := fun i => (μ (s i)).toReal)
    (hsμ : (fun i => (μ (s i)).toReal) =ᶠ[li] m := by rfl) :
    (fun i => (∫ x in s i, f x ∂μ) - m i • f x) =o[li] m 
goal: (fun i => ∫ (x : X) in s i, f x ∂μ - m i • f x) =o[li] m

full_name: ContinuousAt.integral_sub_linear_isLittleO_ae
code: /-- Fundamental theorem of calculus for set integrals, `nhds` version: if `μ` is a locally finite
measure and `f` is an almost everywhere measurable function that is continuous at a point `a`, then
`∫ x in s i, f x ∂μ = μ (s i) • f a + o(μ (s i))` at `li` provided that `s` tends to
`(𝓝 a).smallSets` along `li`. Since `μ (s i)` is an `ℝ≥0∞` number, we use `(μ (s i)).toReal` in
the actual statement.

Often there is a good formula for `(μ (s i)).toReal`, so the formalization can take an optional
argument `m` with this formula and a proof of `(fun i => (μ (s i)).toReal) =ᶠ[li] m`. Without these
arguments, `m i = (μ (s i)).toReal` is used in the output. -/
theorem ContinuousAt.integral_sub_linear_isLittleO_ae [TopologicalSpace X] [OpensMeasurableSpace X]
    {μ : Measure X} [IsLocallyFiniteMeasure μ] {x : X}
    {f : X → E} (hx : ContinuousAt f x) (hfm : StronglyMeasurableAtFilter f (𝓝 x) μ) {s : ι → Set X}
    {li : Filter ι} (hs : Tendsto s li (𝓝 x).smallSets) (m : ι → ℝ := fun i => (μ (s i)).toReal)
    (hsμ : (fun i => (μ (s i)).toReal) =ᶠ[li] m := by rfl) :
    (fun i => (∫ x in s i, f x ∂μ) - m i • f x) =o[li] m 
goal: (fun i => ∫ (x : X) in s i, f x ∂μ - m i • f x) =o[li] m

full_name: ContinuousOn.integral_sub_linear_isLittleO_ae
code: /-- Fundamental theorem of calculus for set integrals, `nhdsWithin` version: if `μ` is a locally
finite measure, `f` is continuous on a measurable set `t`, and `a ∈ t`, then `∫ x in (s i), f x ∂μ =
μ (s i) • f a + o(μ (s i))` at `li` provided that `s i` tends to `(𝓝[t] a).smallSets` along `li`.
Since `μ (s i)` is an `ℝ≥0∞` number, we use `(μ (s i)).toReal` in the actual statement.

Often there is a good formula for `(μ (s i)).toReal`, so the formalization can take an optional
argument `m` with this formula and a proof of `(fun i => (μ (s i)).toReal) =ᶠ[li] m`. Without these
arguments, `m i = (μ (s i)).toReal` is used in the output. -/
theorem ContinuousOn.integral_sub_linear_isLittleO_ae [TopologicalSpace X] [OpensMeasurableSpace X]
    [SecondCountableTopologyEither X E] {μ : Measure X}
    [IsLocallyFiniteMeasure μ] {x : X} {t : Set X} {f : X → E} (hft : ContinuousOn f t) (hx : x ∈ t)
    (ht : MeasurableSet t) {s : ι → Set X} {li : Filter ι} (hs : Tendsto s li (𝓝[t] x).smallSets)
    (m : ι → ℝ := fun i => (μ (s i)).toReal)
    (hsμ : (fun i => (μ (s i)).toReal) =ᶠ[li] m := by rfl) :
    (fun i => (∫ x in s i, f x ∂μ) - m i • f x) =o[li] m 
goal: (fun i => ∫ (x : X) in s i, f x ∂μ - m i • f x) =o[li] m

full_name: ContinuousLinearMap.integral_compLp
code: theorem integral_compLp (L : E →L[𝕜] F) (φ : Lp E p μ) :
    ∫ x, (L.compLp φ) x ∂μ = ∫ x, L (φ x) ∂μ 
goal: ∫ (x : X), ↑↑(L.compLp φ) x ∂μ = ∫ (x : X), L (↑↑φ x) ∂μ

full_name: ContinuousLinearMap.setIntegral_compLp
code: theorem setIntegral_compLp (L : E →L[𝕜] F) (φ : Lp E p μ) {s : Set X} (hs : MeasurableSet s) :
    ∫ x in s, (L.compLp φ) x ∂μ = ∫ x in s, L (φ x) ∂μ 
goal: ∫ (x : X) in s, ↑↑(L.compLp φ) x ∂μ = ∫ (x : X) in s, L (↑↑φ x) ∂μ

full_name: ContinuousLinearMap.continuous_integral_comp_L1
code: theorem continuous_integral_comp_L1 (L : E →L[𝕜] F) :
    Continuous fun φ : X →₁[μ] E => ∫ x : X, L (φ x) ∂μ 
goal: Continuous fun φ => ∫ (x : X), L (↑↑φ x) ∂μ

full_name: ContinuousLinearMap.integral_comp_comm
code: theorem integral_comp_comm [CompleteSpace E] (L : E →L[𝕜] F) {φ : X → E} (φ_int : Integrable φ μ) :
    ∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ) 
goal: ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)

full_name: ContinuousLinearMap.integral_apply
code: theorem integral_apply {H : Type*} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {φ : X → H →L[𝕜] E}
    (φ_int : Integrable φ μ) (v : H) : (∫ x, φ x ∂μ) v = ∫ x, φ x v ∂μ 
goal: (∫ (x : X), φ x ∂μ) v = ∫ (x : X), (φ x) v ∂μ

full_name: ContinuousMultilinearMap.integral_apply
code: theorem _root_.ContinuousMultilinearMap.integral_apply {ι : Type*} [Fintype ι] {M : ι → Type*}
    [∀ i, NormedAddCommGroup (M i)] [∀ i, NormedSpace 𝕜 (M i)]
    {φ : X → ContinuousMultilinearMap 𝕜 M E} (φ_int : Integrable φ μ) (m : ∀ i, M i) :
    (∫ x, φ x ∂μ) m = ∫ x, φ x m ∂μ 
goal: (∫ (x : X), φ x ∂μ) m = ∫ (x : X), (φ x) m ∂μ

full_name: ContinuousLinearMap.integral_comp_comm'
code: theorem integral_comp_comm' (L : E →L[𝕜] F) {K} (hL : AntilipschitzWith K L) (φ : X → E) :
    ∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ) 
goal: ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)

full_name: ContinuousLinearMap.integral_comp_L1_comm
code: theorem integral_comp_L1_comm (L : E →L[𝕜] F) (φ : X →₁[μ] E) :
    ∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ) 
goal: ∫ (x : X), L (↑↑φ x) ∂μ = L (∫ (x : X), ↑↑φ x ∂μ)

full_name: LinearIsometry.integral_comp_comm
code: theorem integral_comp_comm (L : E →ₗᵢ[𝕜] F) (φ : X → E) : ∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ) 
goal: ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)

full_name: ContinuousLinearEquiv.integral_comp_comm
code: theorem integral_comp_comm (L : E ≃L[𝕜] F) (φ : X → E) : ∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ) 
goal: ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)

full_name: integral_ofReal
code: @[norm_cast]
theorem integral_ofReal {f : X → ℝ} : ∫ x, (f x : 𝕜) ∂μ = ↑(∫ x, f x ∂μ) 
goal: ∫ (x : X), ↑(f x) ∂μ = ↑(∫ (x : X), f x ∂μ)

full_name: integral_re
code: theorem integral_re {f : X → 𝕜} (hf : Integrable f μ) :
    ∫ x, RCLike.re (f x) ∂μ = RCLike.re (∫ x, f x ∂μ) 
goal: ∫ (x : X), RCLike.re (f x) ∂μ = RCLike.re (∫ (x : X), f x ∂μ)

full_name: integral_im
code: theorem integral_im {f : X → 𝕜} (hf : Integrable f μ) :
    ∫ x, RCLike.im (f x) ∂μ = RCLike.im (∫ x, f x ∂μ) 
goal: ∫ (x : X), RCLike.im (f x) ∂μ = RCLike.im (∫ (x : X), f x ∂μ)

full_name: integral_conj
code: theorem integral_conj {f : X → 𝕜} : ∫ x, conj (f x) ∂μ = conj (∫ x, f x ∂μ) 
goal: ∫ (x : X), (starRingEnd 𝕜) (f x) ∂μ = (starRingEnd 𝕜) (∫ (x : X), f x ∂μ)

full_name: integral_coe_re_add_coe_im
code: theorem integral_coe_re_add_coe_im {f : X → 𝕜} (hf : Integrable f μ) :
    ∫ x, (re (f x) : 𝕜) ∂μ + (∫ x, (im (f x) : 𝕜) ∂μ) * RCLike.I = ∫ x, f x ∂μ 
goal: ∫ (x : X), ↑(RCLike.re (f x)) ∂μ + (∫ (x : X), ↑(RCLike.im (f x)) ∂μ) * RCLike.I = ∫ (x : X), f x ∂μ

full_name: integral_re_add_im
code: theorem integral_re_add_im {f : X → 𝕜} (hf : Integrable f μ) :
    ((∫ x, RCLike.re (f x) ∂μ : ℝ) : 𝕜) + (∫ x, RCLike.im (f x) ∂μ : ℝ) * RCLike.I =
      ∫ x, f x ∂μ 
goal: ↑(∫ (x : X), RCLike.re (f x) ∂μ) + ↑(∫ (x : X), RCLike.im (f x) ∂μ) * RCLike.I = ∫ (x : X), f x ∂μ

full_name: setIntegral_re_add_im
code: theorem setIntegral_re_add_im {f : X → 𝕜} {i : Set X} (hf : IntegrableOn f i μ) :
    ((∫ x in i, RCLike.re (f x) ∂μ : ℝ) : 𝕜) + (∫ x in i, RCLike.im (f x) ∂μ : ℝ) * RCLike.I =
      ∫ x in i, f x ∂μ 
goal: ↑(∫ (x : X) in i, RCLike.re (f x) ∂μ) + ↑(∫ (x : X) in i, RCLike.im (f x) ∂μ) * RCLike.I = ∫ (x : X) in i, f x ∂μ

full_name: swap_integral
code: lemma swap_integral (f : X → E × F) : (∫ x, f x ∂μ).swap = ∫ x, (f x).swap ∂μ 
goal: (∫ (x : X), f x ∂μ).swap = ∫ (x : X), (f x).swap ∂μ

full_name: fst_integral
code: theorem fst_integral [CompleteSpace F] {f : X → E × F} (hf : Integrable f μ) :
    (∫ x, f x ∂μ).1 = ∫ x, (f x).1 ∂μ 
goal: (∫ (x : X), f x ∂μ).1 = ∫ (x : X), (f x).1 ∂μ

full_name: snd_integral
code: theorem snd_integral [CompleteSpace E] {f : X → E × F} (hf : Integrable f μ) :
    (∫ x, f x ∂μ).2 = ∫ x, (f x).2 ∂μ 
goal: (∫ (x : X), f x ∂μ).2 = ∫ (x : X), (f x).2 ∂μ

full_name: integral_pair
code: theorem integral_pair [CompleteSpace E] [CompleteSpace F] {f : X → E} {g : X → F}
    (hf : Integrable f μ) (hg : Integrable g μ) :
    ∫ x, (f x, g x) ∂μ = (∫ x, f x ∂μ, ∫ x, g x ∂μ) 
goal: ∫ (x : X), (f x, g x) ∂μ = (∫ (x : X), f x ∂μ, ∫ (x : X), g x ∂μ)

full_name: integral_smul_const
code: theorem integral_smul_const {𝕜 : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] [CompleteSpace E]
    (f : X → 𝕜) (c : E) :
    ∫ x, f x • c ∂μ = (∫ x, f x ∂μ) • c 
goal: ∫ (x : X), f x • c ∂μ = (∫ (x : X), f x ∂μ) • c

full_name: integral_withDensity_eq_integral_smul
code: theorem integral_withDensity_eq_integral_smul {f : X → ℝ≥0} (f_meas : Measurable f) (g : X → E) :
    ∫ x, g x ∂μ.withDensity (fun x => f x) = ∫ x, f x • g x ∂μ 
goal: (∫ (x : X), g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • g x ∂μ

full_name: integral_withDensity_eq_integral_smul₀
code: theorem integral_withDensity_eq_integral_smul₀ {f : X → ℝ≥0} (hf : AEMeasurable f μ) (g : X → E) :
    ∫ x, g x ∂μ.withDensity (fun x => f x) = ∫ x, f x • g x ∂μ 
goal: (∫ (x : X), g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X), f x • g x ∂μ

full_name: setIntegral_withDensity_eq_setIntegral_smul
code: theorem setIntegral_withDensity_eq_setIntegral_smul {f : X → ℝ≥0} (f_meas : Measurable f)
    (g : X → E) {s : Set X} (hs : MeasurableSet s) :
    ∫ x in s, g x ∂μ.withDensity (fun x => f x) = ∫ x in s, f x • g x ∂μ 
goal: (∫ (x : X) in s, g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X) in s, f x • g x ∂μ

full_name: setIntegral_withDensity_eq_setIntegral_smul₀
code: theorem setIntegral_withDensity_eq_setIntegral_smul₀ {f : X → ℝ≥0} {s : Set X}
    (hf : AEMeasurable f (μ.restrict s)) (g : X → E) (hs : MeasurableSet s) :
    ∫ x in s, g x ∂μ.withDensity (fun x => f x) = ∫ x in s, f x • g x ∂μ 
goal: (∫ (x : X) in s, g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X) in s, f x • g x ∂μ

full_name: setIntegral_withDensity_eq_setIntegral_smul₀'
code: theorem setIntegral_withDensity_eq_setIntegral_smul₀' [SFinite μ] {f : X → ℝ≥0} (s : Set X)
    (hf : AEMeasurable f (μ.restrict s)) (g : X → E)  :
    ∫ x in s, g x ∂μ.withDensity (fun x => f x) = ∫ x in s, f x • g x ∂μ 
goal: (∫ (x : X) in s, g x ∂μ.withDensity fun x => ↑(f x)) = ∫ (x : X) in s, f x • g x ∂μ

full_name: measure_le_lintegral_thickenedIndicatorAux
code: theorem measure_le_lintegral_thickenedIndicatorAux (μ : Measure X) {E : Set X}
    (E_mble : MeasurableSet E) (δ : ℝ) : μ E ≤ ∫⁻ x, (thickenedIndicatorAux δ E x : ℝ≥0∞) ∂μ 
goal: μ E ≤ ∫⁻ (x : X), thickenedIndicatorAux δ E x ∂μ

full_name: measure_le_lintegral_thickenedIndicator
code: theorem measure_le_lintegral_thickenedIndicator (μ : Measure X) {E : Set X}
    (E_mble : MeasurableSet E) {δ : ℝ} (δ_pos : 0 < δ) :
    μ E ≤ ∫⁻ x, (thickenedIndicator δ_pos E x : ℝ≥0∞) ∂μ 
goal: μ E ≤ ∫⁻ (x : X), ↑((thickenedIndicator δ_pos E) x) ∂μ

full_name: MeasureTheory.Integrable.simpleFunc_mul
code: theorem Integrable.simpleFunc_mul (g : SimpleFunc X ℝ) (hf : Integrable f μ) :
    Integrable (⇑g * f) μ 
goal: MeasureTheory.Integrable (↑g * f) μ

full_name: MeasureTheory.Integrable.simpleFunc_mul'
code: theorem Integrable.simpleFunc_mul' (hm : m ≤ m0) (g : @SimpleFunc X m ℝ) (hf : Integrable f μ) :
    Integrable (⇑g * f) μ 
goal: MeasureTheory.Integrable (↑g * f) μ

full_name: continuous_parametric_integral_of_continuous
code: /-- The parametric integral over a continuous function on a compact set is continuous,
  under mild assumptions on the topologies involved. -/
theorem continuous_parametric_integral_of_continuous
    [FirstCountableTopology X] [LocallyCompactSpace X]
    [OpensMeasurableSpace Y] [SecondCountableTopologyEither Y E] [IsLocallyFiniteMeasure μ]
    {f : X → Y → E} (hf : Continuous f.uncurry) {s : Set Y} (hs : IsCompact s) :
    Continuous (∫ y in s, f · y ∂μ) 
goal: Continuous fun x => ∫ (y : Y) in s, f x y ∂μ

full_name: continuousOn_integral_bilinear_of_locally_integrable_of_compact_support
code: /-- Consider a parameterized integral `x ↦ ∫ y, L (g y) (f x y)` where `L` is bilinear,
`g` is locally integrable and `f` is continuous and uniformly compactly supported. Then the
integral depends continuously on `x`. -/
lemma continuousOn_integral_bilinear_of_locally_integrable_of_compact_support
    [NormedSpace 𝕜 E] (L : F →L[𝕜] G →L[𝕜] E)
    {f : X → Y → G} {s : Set X} {k : Set Y} {g : Y → F}
    (hk : IsCompact k) (hf : ContinuousOn f.uncurry (s ×ˢ univ))
    (hfs : ∀ p, ∀ x, p ∈ s → x ∉ k → f p x = 0) (hg : IntegrableOn g k μ) :
    ContinuousOn (fun x ↦ ∫ y, L (g y) (f x y) ∂μ) s 
goal: ContinuousOn (fun x => ∫ (y : Y), (L (g y)) (f x y) ∂μ) s

full_name: continuousOn_integral_of_compact_support
code: /-- Consider a parameterized integral `x ↦ ∫ y, f x y` where `f` is continuous and uniformly
compactly supported. Then the integral depends continuously on `x`. -/
lemma continuousOn_integral_of_compact_support
    {f : X → Y → E} {s : Set X} {k : Set Y} [IsFiniteMeasureOnCompacts μ]
    (hk : IsCompact k) (hf : ContinuousOn f.uncurry (s ×ˢ univ))
    (hfs : ∀ p, ∀ x, p ∈ s → x ∉ k → f p x = 0) :
    ContinuousOn (fun x ↦ ∫ y, f x y ∂μ) s 
goal: ContinuousOn (fun x => ∫ (y : Y), f x y ∂μ) s