ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Function.Periodic.sInf_add_zsmul_le_integral_of_pos
theorem sInf_add_zsmul_le_integral_of_pos (hT : 0 < T) (t : ℝ) :
    (sInf ((fun t => ∫ x in (0)..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ x in (0)..T, g x) ≤
      ∫ x in (0)..t, g x 
Arguments: {T : ℝ} {g : ℝ → ℝ} (hg : Function.Periodic g T) (h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable g MeasureTheory.volume t₁ t₂) (hT : 0 < T) (t : ℝ)
Goal: sInf ((fun t => ∫ (x : ℝ) in 0 ..t, g x) '' Set.Icc 0 T) + ⌊t / T⌋ • ∫ (x : ℝ) in 0 ..T, g x ≤ ∫ (x : ℝ) in 0 ..t, g x
-- Function.Periodic.integral_le_sSup_add_zsmul_of_pos
theorem integral_le_sSup_add_zsmul_of_pos (hT : 0 < T) (t : ℝ) :
    (∫ x in (0)..t, g x) ≤
      sSup ((fun t => ∫ x in (0)..t, g x) '' Icc 0 T) + ⌊t / T⌋ • ∫ x in (0)..T, g x 
Arguments: {T : ℝ} {g : ℝ → ℝ} (hg : Function.Periodic g T) (h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable g MeasureTheory.volume t₁ t₂) (hT : 0 < T) (t : ℝ)
Goal: ∫ (x : ℝ) in 0 ..t, g x ≤ sSup ((fun t => ∫ (x : ℝ) in 0 ..t, g x) '' Set.Icc 0 T) + ⌊t / T⌋ • ∫ (x : ℝ) in 0 ..T, g x
-- Function.Periodic.tendsto_atTop_intervalIntegral_of_pos
theorem tendsto_atTop_intervalIntegral_of_pos (h₀ : 0 < ∫ x in (0)..T, g x) (hT : 0 < T) :
    Tendsto (fun t => ∫ x in (0)..t, g x) atTop atTop 
Arguments: {T : ℝ} {g : ℝ → ℝ} (hg : Function.Periodic g T) (h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable g MeasureTheory.volume t₁ t₂) (h₀ : 0 < ∫ (x : ℝ) in 0 ..T, g x) (hT : 0 < T)
Goal: Filter.Tendsto (fun t => ∫ (x : ℝ) in 0 ..t, g x) Filter.atTop Filter.atTop
-- Function.Periodic.tendsto_atBot_intervalIntegral_of_pos
theorem tendsto_atBot_intervalIntegral_of_pos (h₀ : 0 < ∫ x in (0)..T, g x) (hT : 0 < T) :
    Tendsto (fun t => ∫ x in (0)..t, g x) atBot atBot 
Arguments: {T : ℝ} {g : ℝ → ℝ} (hg : Function.Periodic g T) (h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable g MeasureTheory.volume t₁ t₂) (h₀ : 0 < ∫ (x : ℝ) in 0 ..T, g x) (hT : 0 < T)
Goal: Filter.Tendsto (fun t => ∫ (x : ℝ) in 0 ..t, g x) Filter.atBot Filter.atBot
-- Function.Periodic.tendsto_atTop_intervalIntegral_of_pos'
theorem tendsto_atTop_intervalIntegral_of_pos' (h₀ : ∀ x, 0 < g x) (hT : 0 < T) :
    Tendsto (fun t => ∫ x in (0)..t, g x) atTop atTop 
Arguments: {T : ℝ} {g : ℝ → ℝ} (hg : Function.Periodic g T) (h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable g MeasureTheory.volume t₁ t₂) (h₀ : ∀ (x : ℝ), 0 < g x) (hT : 0 < T)
Goal: Filter.Tendsto (fun t => ∫ (x : ℝ) in 0 ..t, g x) Filter.atTop Filter.atTop
-- Function.Periodic.tendsto_atBot_intervalIntegral_of_pos'
theorem tendsto_atBot_intervalIntegral_of_pos' (h₀ : ∀ x, 0 < g x) (hT : 0 < T) :
    Tendsto (fun t => ∫ x in (0)..t, g x) atBot atBot 
Arguments: {T : ℝ} {g : ℝ → ℝ} (hg : Function.Periodic g T) (h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable g MeasureTheory.volume t₁ t₂) (h₀ : ∀ (x : ℝ), 0 < g x) (hT : 0 < T)
Goal: Filter.Tendsto (fun t => ∫ (x : ℝ) in 0 ..t, g x) Filter.atBot Filter.atBot
-- unitInterval.volume_def
theorem volume_def : (volume : Measure I) = volume.comap Subtype.val 
Goal: MeasureTheory.volume = MeasureTheory.Measure.comap Subtype.val MeasureTheory.volume
-- unitInterval.measurable_symm
@[measurability]
theorem measurable_symm : Measurable symm 
Goal: Measurable σ
-- integral_rpow_mul_exp_neg_rpow
theorem integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 0 < p) (hq : - 1 < q) :
    ∫ x in Ioi (0 : ℝ), x ^ q * exp (- x ^ p) = (1 / p) * Gamma ((q + 1) / p) 
Arguments: {p q : ℝ} (hp : 0 < p) (hq : -1 < q)
Goal: ∫ (x : ℝ) in Set.Ioi 0, x ^ q * rexp (-x ^ p) = 1 / p * Real.Gamma ((q + 1) / p)
-- integral_rpow_mul_exp_neg_mul_rpow
theorem integral_rpow_mul_exp_neg_mul_rpow {p q b : ℝ} (hp : 0 < p) (hq : - 1 < q) (hb : 0 < b) :
    ∫ x in Ioi (0 : ℝ), x ^ q * exp (- b * x ^ p) =
      b ^ (-(q + 1) / p) * (1 / p) * Gamma ((q + 1) / p) 
Arguments: {p q b : ℝ} (hp : 0 < p) (hq : -1 < q) (hb : 0 < b)
Goal: ∫ (x : ℝ) in Set.Ioi 0, x ^ q * rexp (-b * x ^ p) = b ^ (-(q + 1) / p) * (1 / p) * Real.Gamma ((q + 1) / p)
-- integral_exp_neg_rpow
theorem integral_exp_neg_rpow {p : ℝ} (hp : 0 < p) :
    ∫ x in Ioi (0 : ℝ), exp (- x ^ p) = Gamma (1 / p + 1) 
Arguments: {p : ℝ} (hp : 0 < p)
Goal: ∫ (x : ℝ) in Set.Ioi 0, rexp (-x ^ p) = Real.Gamma (1 / p + 1)
-- integral_exp_neg_mul_rpow
theorem integral_exp_neg_mul_rpow {p b : ℝ} (hp : 0 < p) (hb : 0 < b) :
    ∫ x in Ioi (0 : ℝ), exp (- b * x ^ p) = b ^ (- 1 / p) * Gamma (1 / p + 1) 
Arguments: {p b : ℝ} (hp : 0 < p) (hb : 0 < b)
Goal: ∫ (x : ℝ) in Set.Ioi 0, rexp (-b * x ^ p) = b ^ (-1 / p) * Real.Gamma (1 / p + 1)
-- Complex.integral_rpow_mul_exp_neg_rpow
theorem Complex.integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 1 ≤ p) (hq : - 2 < q) :
    ∫ x : ℂ, ‖x‖ ^ q * rexp (- ‖x‖ ^ p) = (2 * π / p) * Real.Gamma ((q + 2) / p) 
Arguments: {p q : ℝ} (hp : 1 ≤ p) (hq : -2 < q)
Goal: ∫ (x : ℂ), ‖x‖ ^ q * rexp (-‖x‖ ^ p) = 2 * π / p * Real.Gamma ((q + 2) / p)
-- Complex.integral_rpow_mul_exp_neg_mul_rpow
theorem Complex.integral_rpow_mul_exp_neg_mul_rpow {p q b : ℝ} (hp : 1 ≤ p) (hq : - 2 < q)
    (hb : 0 < b) :
    ∫ x : ℂ, ‖x‖ ^ q * rexp (- b * ‖x‖ ^ p) = (2 * π / p) *
      b ^ (-(q + 2) / p) * Real.Gamma ((q + 2) / p) 
Arguments: {p q b : ℝ} (hp : 1 ≤ p) (hq : -2 < q) (hb : 0 < b)
Goal: ∫ (x : ℂ), ‖x‖ ^ q * rexp (-b * ‖x‖ ^ p) = 2 * π / p * b ^ (-(q + 2) / p) * Real.Gamma ((q + 2) / p)
-- Complex.integral_exp_neg_rpow
theorem Complex.integral_exp_neg_rpow {p : ℝ} (hp : 1 ≤ p) :
    ∫ x : ℂ, rexp (- ‖x‖ ^ p) = π * Real.Gamma (2 / p + 1) 
Arguments: {p : ℝ} (hp : 1 ≤ p)
Goal: ∫ (x : ℂ), rexp (-‖x‖ ^ p) = π * Real.Gamma (2 / p + 1)
-- Complex.integral_exp_neg_mul_rpow
theorem Complex.integral_exp_neg_mul_rpow {p b : ℝ} (hp : 1 ≤ p) (hb : 0 < b) :
    ∫ x : ℂ, rexp (- b * ‖x‖ ^ p) = π * b ^ (-2 / p) * Real.Gamma (2 / p + 1) 
Arguments: {p b : ℝ} (hp : 1 ≤ p) (hb : 0 < b)
Goal: ∫ (x : ℂ), rexp (-b * ‖x‖ ^ p) = π * b ^ (-2 / p) * Real.Gamma (2 / p + 1)
-- MeasureTheory.IsProjectiveMeasureFamily.measure_univ_eq_of_subset
lemma measure_univ_eq_of_subset (hP : IsProjectiveMeasureFamily P) (hJI : J ⊆ I) :
    P I univ = P J univ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {P : (J : Finset ι) → MeasureTheory.Measure ((j : { x // x ∈ J }) → α ↑j)} {I J : Finset ι} (hP : MeasureTheory.IsProjectiveMeasureFamily P) (hJI : J ⊆ I)
Goal: (P I) Set.univ = (P J) Set.univ
-- MeasureTheory.IsProjectiveMeasureFamily.measure_univ_eq
lemma measure_univ_eq (hP : IsProjectiveMeasureFamily P) (I J : Finset ι) :
    P I univ = P J univ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {P : (J : Finset ι) → MeasureTheory.Measure ((j : { x // x ∈ J }) → α ↑j)} (hP : MeasureTheory.IsProjectiveMeasureFamily P) (I J : Finset ι)
Goal: (P I) Set.univ = (P J) Set.univ
-- MeasureTheory.IsProjectiveMeasureFamily.congr_cylinder_of_subset
lemma congr_cylinder_of_subset (hP : IsProjectiveMeasureFamily P)
    {S : Set (∀ i : I, α i)} {T : Set (∀ i : J, α i)} (hT : MeasurableSet T)
    (h_eq : cylinder I S = cylinder J T) (hJI : J ⊆ I) :
    P I S = P J T 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {P : (J : Finset ι) → MeasureTheory.Measure ((j : { x // x ∈ J }) → α ↑j)} {I J : Finset ι} (hP : MeasureTheory.IsProjectiveMeasureFamily P) {S : Set ((i : { x // x ∈ I }) → α ↑i)} {T : Set ((i : { x // x ∈ J }) → α ↑i)} (hT : MeasurableSet T) (h_eq : MeasureTheory.cylinder I S = MeasureTheory.cylinder J T) (hJI : J ⊆ I)
Goal: (P I) S = (P J) T
-- MeasureTheory.IsProjectiveMeasureFamily.congr_cylinder
lemma congr_cylinder (hP : IsProjectiveMeasureFamily P)
    {S : Set (∀ i : I, α i)} {T : Set (∀ i : J, α i)} (hS : MeasurableSet S) (hT : MeasurableSet T)
    (h_eq : cylinder I S = cylinder J T) :
    P I S = P J T 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {P : (J : Finset ι) → MeasureTheory.Measure ((j : { x // x ∈ J }) → α ↑j)} {I J : Finset ι} (hP : MeasureTheory.IsProjectiveMeasureFamily P) {S : Set ((i : { x // x ∈ I }) → α ↑i)} {T : Set ((i : { x // x ∈ J }) → α ↑i)} (hS : MeasurableSet S) (hT : MeasurableSet T) (h_eq : MeasureTheory.cylinder I S = MeasureTheory.cylinder J T)
Goal: (P I) S = (P J) T
-- MeasureTheory.IsProjectiveLimit.measure_cylinder
lemma measure_cylinder (h : IsProjectiveLimit μ P)
    (I : Finset ι) {s : Set (∀ i : I, α i)} (hs : MeasurableSet s) :
    μ (cylinder I s) = P I s 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {P : (J : Finset ι) → MeasureTheory.Measure ((j : { x // x ∈ J }) → α ↑j)} {μ : MeasureTheory.Measure ((i : ι) → α i)} (h : MeasureTheory.IsProjectiveLimit μ P) (I : Finset ι) {s : Set ((i : { x // x ∈ I }) → α ↑i)} (hs : MeasurableSet s)
Goal: μ (MeasureTheory.cylinder I s) = (P I) s
-- MeasureTheory.IsProjectiveLimit.measure_univ_eq
lemma measure_univ_eq (hμ : IsProjectiveLimit μ P) (I : Finset ι) :
    μ univ = P I univ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {P : (J : Finset ι) → MeasureTheory.Measure ((j : { x // x ∈ J }) → α ↑j)} {μ : MeasureTheory.Measure ((i : ι) → α i)} (hμ : MeasureTheory.IsProjectiveLimit μ P) (I : Finset ι)
Goal: μ Set.univ = (P I) Set.univ
-- MeasureTheory.IsProjectiveLimit.isFiniteMeasure
lemma isFiniteMeasure [∀ i, IsFiniteMeasure (P i)] (hμ : IsProjectiveLimit μ P) :
    IsFiniteMeasure μ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {P : (J : Finset ι) → MeasureTheory.Measure ((j : { x // x ∈ J }) → α ↑j)} {μ : MeasureTheory.Measure ((i : ι) → α i)} [∀ (i : Finset ι), MeasureTheory.IsFiniteMeasure (P i)] (hμ : MeasureTheory.IsProjectiveLimit μ P)
Goal: MeasureTheory.IsFiniteMeasure μ
-- MeasureTheory.IsProjectiveLimit.isProbabilityMeasure
lemma isProbabilityMeasure [∀ i, IsProbabilityMeasure (P i)] (hμ : IsProjectiveLimit μ P) :
    IsProbabilityMeasure μ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {P : (J : Finset ι) → MeasureTheory.Measure ((j : { x // x ∈ J }) → α ↑j)} {μ : MeasureTheory.Measure ((i : ι) → α i)} [∀ (i : Finset ι), MeasureTheory.IsProbabilityMeasure (P i)] (hμ : MeasureTheory.IsProjectiveLimit μ P)
Goal: MeasureTheory.IsProbabilityMeasure μ
-- MeasureTheory.IsProjectiveLimit.measure_univ_unique
lemma measure_univ_unique (hμ : IsProjectiveLimit μ P) (hν : IsProjectiveLimit ν P) :
    μ univ = ν univ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {P : (J : Finset ι) → MeasureTheory.Measure ((j : { x // x ∈ J }) → α ↑j)} {μ ν : MeasureTheory.Measure ((i : ι) → α i)} (hμ : MeasureTheory.IsProjectiveLimit μ P) (hν : MeasureTheory.IsProjectiveLimit ν P)
Goal: μ Set.univ = ν Set.univ
-- MeasureTheory.IsProjectiveLimit.unique
theorem unique [∀ i, IsFiniteMeasure (P i)]
    (hμ : IsProjectiveLimit μ P) (hν : IsProjectiveLimit ν P) :
    μ = ν 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → MeasurableSpace (α i)] {P : (J : Finset ι) → MeasureTheory.Measure ((j : { x // x ∈ J }) → α ↑j)} {μ ν : MeasureTheory.Measure ((i : ι) → α i)} [∀ (i : Finset ι), MeasureTheory.IsFiniteMeasure (P i)] (hμ : MeasureTheory.IsProjectiveLimit μ P) (hν : MeasureTheory.IsProjectiveLimit ν P)
Goal: μ = ν
-- stronglyMeasurableAt_bot
@[simp]
theorem stronglyMeasurableAt_bot {f : α → β} : StronglyMeasurableAtFilter f ⊥ μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] {μ : MeasureTheory.Measure α} {f : α → β}
Goal: StronglyMeasurableAtFilter f ⊥ μ
-- StronglyMeasurableAtFilter.eventually
protected theorem StronglyMeasurableAtFilter.eventually (h : StronglyMeasurableAtFilter f l μ) :
    ∀ᶠ s in l.smallSets, AEStronglyMeasurable f (μ.restrict s) 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] {l : Filter α} {f : α → β} {μ : MeasureTheory.Measure α} (h : StronglyMeasurableAtFilter f l μ)
Goal: ∀ᶠ (s : Set α) in l.smallSets, MeasureTheory.AEStronglyMeasurable f (μ.restrict s)
-- StronglyMeasurableAtFilter.filter_mono
protected theorem StronglyMeasurableAtFilter.filter_mono (h : StronglyMeasurableAtFilter f l μ)
    (h' : l' ≤ l) : StronglyMeasurableAtFilter f l' μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] {l l' : Filter α} {f : α → β} {μ : MeasureTheory.Measure α} (h : StronglyMeasurableAtFilter f l μ) (h' : l' ≤ l)
Goal: StronglyMeasurableAtFilter f l' μ
-- MeasureTheory.AEStronglyMeasurable.stronglyMeasurableAtFilter
protected theorem MeasureTheory.AEStronglyMeasurable.stronglyMeasurableAtFilter
    (h : AEStronglyMeasurable f μ) : StronglyMeasurableAtFilter f l μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] {l : Filter α} {f : α → β} {μ : MeasureTheory.Measure α} (h : MeasureTheory.AEStronglyMeasurable f μ)
Goal: StronglyMeasurableAtFilter f l μ
-- AeStronglyMeasurable.stronglyMeasurableAtFilter_of_mem
theorem AeStronglyMeasurable.stronglyMeasurableAtFilter_of_mem {s}
    (h : AEStronglyMeasurable f (μ.restrict s)) (hl : s ∈ l) : StronglyMeasurableAtFilter f l μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] {l : Filter α} {f : α → β} {μ : MeasureTheory.Measure α} {s : Set α} (h : MeasureTheory.AEStronglyMeasurable f (μ.restrict s)) (hl : s ∈ l)
Goal: StronglyMeasurableAtFilter f l μ
-- MeasureTheory.StronglyMeasurable.stronglyMeasurableAtFilter
protected theorem MeasureTheory.StronglyMeasurable.stronglyMeasurableAtFilter
    (h : StronglyMeasurable f) : StronglyMeasurableAtFilter f l μ 
Arguments: {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [TopologicalSpace β] {l : Filter α} {f : α → β} {μ : MeasureTheory.Measure α} (h : MeasureTheory.StronglyMeasurable f)
Goal: StronglyMeasurableAtFilter f l μ
-- MeasureTheory.hasFiniteIntegral_restrict_of_bounded
theorem hasFiniteIntegral_restrict_of_bounded [NormedAddCommGroup E] {f : α → E} {s : Set α}
    {μ : Measure α} {C} (hs : μ s < ∞) (hf : ∀ᵐ x ∂μ.restrict s, ‖f x‖ ≤ C) :
    HasFiniteIntegral f (μ.restrict s) 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ : MeasureTheory.Measure α} {C : ℝ} (hs : μ s < ⊤) (hf : ∀ᵐ (x : α) ∂μ.restrict s, ‖f x‖ ≤ C)
Goal: MeasureTheory.HasFiniteIntegral f (μ.restrict s)
-- MeasureTheory.IntegrableOn.integrable
theorem IntegrableOn.integrable (h : IntegrableOn f s μ) : Integrable f (μ.restrict s) 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f s μ)
Goal: MeasureTheory.Integrable f (μ.restrict s)
-- MeasureTheory.integrableOn_empty
@[simp]
theorem integrableOn_empty : IntegrableOn f ∅ μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.IntegrableOn f ∅ μ
-- MeasureTheory.integrableOn_univ
@[simp]
theorem integrableOn_univ : IntegrableOn f univ μ ↔ Integrable f μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.IntegrableOn f Set.univ μ ↔ MeasureTheory.Integrable f μ
-- MeasureTheory.integrableOn_zero
theorem integrableOn_zero : IntegrableOn (fun _ => (0 : E)) s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.IntegrableOn (fun x => 0) s μ
-- MeasureTheory.integrableOn_const
@[simp]
theorem integrableOn_const {C : E} : IntegrableOn (fun _ => C) s μ ↔ C = 0 ∨ μ s < ∞ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {s : Set α} {μ : MeasureTheory.Measure α} {C : E}
Goal: MeasureTheory.IntegrableOn (fun x => C) s μ ↔ C = 0 ∨ μ s < ⊤
-- MeasureTheory.IntegrableOn.mono
theorem IntegrableOn.mono (h : IntegrableOn f t ν) (hs : s ⊆ t) (hμ : μ ≤ ν) : IntegrableOn f s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ ν : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f t ν) (hs : s ⊆ t) (hμ : μ ≤ ν)
Goal: MeasureTheory.IntegrableOn f s μ
-- MeasureTheory.IntegrableOn.mono_set
theorem IntegrableOn.mono_set (h : IntegrableOn f t μ) (hst : s ⊆ t) : IntegrableOn f s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f t μ) (hst : s ⊆ t)
Goal: MeasureTheory.IntegrableOn f s μ
-- MeasureTheory.IntegrableOn.mono_measure
theorem IntegrableOn.mono_measure (h : IntegrableOn f s ν) (hμ : μ ≤ ν) : IntegrableOn f s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ ν : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f s ν) (hμ : μ ≤ ν)
Goal: MeasureTheory.IntegrableOn f s μ
-- MeasureTheory.IntegrableOn.mono_set_ae
theorem IntegrableOn.mono_set_ae (h : IntegrableOn f t μ) (hst : s ≤ᵐ[μ] t) : IntegrableOn f s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f t μ) (hst : s ≤ᵐ[μ] t)
Goal: MeasureTheory.IntegrableOn f s μ
-- MeasureTheory.IntegrableOn.congr_set_ae
theorem IntegrableOn.congr_set_ae (h : IntegrableOn f t μ) (hst : s =ᵐ[μ] t) : IntegrableOn f s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f t μ) (hst : s =ᵐ[μ] t)
Goal: MeasureTheory.IntegrableOn f s μ
-- MeasureTheory.IntegrableOn.congr_fun_ae
theorem IntegrableOn.congr_fun_ae (h : IntegrableOn f s μ) (hst : f =ᵐ[μ.restrict s] g) :
    IntegrableOn g s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f g : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f s μ) (hst : f =ᵐ[μ.restrict s] g)
Goal: MeasureTheory.IntegrableOn g s μ
-- MeasureTheory.integrableOn_congr_fun_ae
theorem integrableOn_congr_fun_ae (hst : f =ᵐ[μ.restrict s] g) :
    IntegrableOn f s μ ↔ IntegrableOn g s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f g : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (hst : f =ᵐ[μ.restrict s] g)
Goal: MeasureTheory.IntegrableOn f s μ ↔ MeasureTheory.IntegrableOn g s μ
-- MeasureTheory.IntegrableOn.congr_fun
theorem IntegrableOn.congr_fun (h : IntegrableOn f s μ) (hst : EqOn f g s) (hs : MeasurableSet s) :
    IntegrableOn g s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f g : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f s μ) (hst : Set.EqOn f g s) (hs : MeasurableSet s)
Goal: MeasureTheory.IntegrableOn g s μ
-- MeasureTheory.integrableOn_congr_fun
theorem integrableOn_congr_fun (hst : EqOn f g s) (hs : MeasurableSet s) :
    IntegrableOn f s μ ↔ IntegrableOn g s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f g : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (hst : Set.EqOn f g s) (hs : MeasurableSet s)
Goal: MeasureTheory.IntegrableOn f s μ ↔ MeasureTheory.IntegrableOn g s μ
-- MeasureTheory.Integrable.integrableOn
theorem Integrable.integrableOn (h : Integrable f μ) : IntegrableOn f s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.Integrable f μ)
Goal: MeasureTheory.IntegrableOn f s μ
-- MeasureTheory.IntegrableOn.restrict
theorem IntegrableOn.restrict (h : IntegrableOn f s μ) (hs : MeasurableSet s) :
    IntegrableOn f s (μ.restrict t) 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f s μ) (hs : MeasurableSet s)
Goal: MeasureTheory.IntegrableOn f s (μ.restrict t)
-- MeasureTheory.IntegrableOn.inter_of_restrict
theorem IntegrableOn.inter_of_restrict (h : IntegrableOn f s (μ.restrict t)) :
    IntegrableOn f (s ∩ t) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f s (μ.restrict t))
Goal: MeasureTheory.IntegrableOn f (s ∩ t) μ
-- MeasureTheory.Integrable.piecewise
lemma Integrable.piecewise [DecidablePred (· ∈ s)]
    (hs : MeasurableSet s) (hf : IntegrableOn f s μ) (hg : IntegrableOn g sᶜ μ) :
    Integrable (s.piecewise f g) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f g : α → E} {s : Set α} {μ : MeasureTheory.Measure α} [DecidablePred fun x => x ∈ s] (hs : MeasurableSet s) (hf : MeasureTheory.IntegrableOn f s μ) (hg : MeasureTheory.IntegrableOn g sᶜ μ)
Goal: MeasureTheory.Integrable (s.piecewise f g) μ
-- MeasureTheory.IntegrableOn.left_of_union
theorem IntegrableOn.left_of_union (h : IntegrableOn f (s ∪ t) μ) : IntegrableOn f s μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f (s ∪ t) μ)
Goal: MeasureTheory.IntegrableOn f s μ
-- MeasureTheory.IntegrableOn.right_of_union
theorem IntegrableOn.right_of_union (h : IntegrableOn f (s ∪ t) μ) : IntegrableOn f t μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ : MeasureTheory.Measure α} (h : MeasureTheory.IntegrableOn f (s ∪ t) μ)
Goal: MeasureTheory.IntegrableOn f t μ
-- MeasureTheory.IntegrableOn.union
theorem IntegrableOn.union (hs : IntegrableOn f s μ) (ht : IntegrableOn f t μ) :
    IntegrableOn f (s ∪ t) μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ : MeasureTheory.Measure α} (hs : MeasureTheory.IntegrableOn f s μ) (ht : MeasureTheory.IntegrableOn f t μ)
Goal: MeasureTheory.IntegrableOn f (s ∪ t) μ
-- MeasureTheory.integrableOn_union
@[simp]
theorem integrableOn_union : IntegrableOn f (s ∪ t) μ ↔ IntegrableOn f s μ ∧ IntegrableOn f t μ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {s t : Set α} {μ : MeasureTheory.Measure α}
Goal: MeasureTheory.IntegrableOn f (s ∪ t) μ ↔ MeasureTheory.IntegrableOn f s μ ∧ MeasureTheory.IntegrableOn f t μ
-- MeasureTheory.integrableOn_singleton_iff
@[simp]
theorem integrableOn_singleton_iff {x : α} [MeasurableSingletonClass α] :
    IntegrableOn f {x} μ ↔ f x = 0 ∨ μ {x} < ∞ 
Arguments: {α : Type u_1} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {x : α} [MeasurableSingletonClass α]
Goal: MeasureTheory.IntegrableOn f {x} μ ↔ f x = 0 ∨ μ {x} < ⊤
-- MeasureTheory.integrableOn_finite_biUnion
@[simp]
theorem integrableOn_finite_biUnion {s : Set β} (hs : s.Finite) {t : β → Set α} :
    IntegrableOn f (⋃ i ∈ s, t i) μ ↔ ∀ i ∈ s, IntegrableOn f (t i) μ 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {s : Set β} (hs : s.Finite) {t : β → Set α}
Goal: MeasureTheory.IntegrableOn f (⋃ i ∈ s, t i) μ ↔ ∀ i ∈ s, MeasureTheory.IntegrableOn f (t i) μ
-- MeasureTheory.integrableOn_finset_iUnion
@[simp]
theorem integrableOn_finset_iUnion {s : Finset β} {t : β → Set α} :
    IntegrableOn f (⋃ i ∈ s, t i) μ ↔ ∀ i ∈ s, IntegrableOn f (t i) μ 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} {s : Finset β} {t : β → Set α}
Goal: MeasureTheory.IntegrableOn f (⋃ i ∈ s, t i) μ ↔ ∀ i ∈ s, MeasureTheory.IntegrableOn f (t i) μ
-- MeasureTheory.integrableOn_finite_iUnion
@[simp]
theorem integrableOn_finite_iUnion [Finite β] {t : β → Set α} :
    IntegrableOn f (⋃ i, t i) μ ↔ ∀ i, IntegrableOn f (t i) μ 
Arguments: {α : Type u_1} {β : Type u_2} {E : Type u_3} [MeasurableSpace α] [NormedAddCommGroup E] {f : α → E} {μ : MeasureTheory.Measure α} [Finite β] {t : β → Set α}
Goal: MeasureTheory.IntegrableOn f (⋃ i, t i) μ ↔ ∀ (i : β), MeasureTheory.IntegrableOn f (t i) μ