ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.exists_lt_lowerSemicontinuous_integral_lt
theorem exists_lt_lowerSemicontinuous_integral_lt [SigmaFinite μ] (f : α → ℝ) (hf : Integrable f μ)
    {ε : ℝ} (εpos : 0 < ε) :
    ∃ g : α → EReal,
      (∀ x, (f x : EReal) < g x) ∧
      LowerSemicontinuous g ∧
      Integrable (fun x => EReal.toReal (g x)) μ ∧
      (∀ᵐ x ∂μ, g x < ⊤) ∧ (∫ x, EReal.toReal (g x) ∂μ) < (∫ x, f x ∂μ) + ε 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] {μ : MeasureTheory.Measure α} [μ.WeaklyRegular] [MeasureTheory.SigmaFinite μ] (f : α → ℝ) (hf : MeasureTheory.Integrable f μ) {ε : ℝ} (εpos : 0 < ε)
Goal: ∃ g,
  (∀ (x : α), ↑(f x) < g x) ∧
    LowerSemicontinuous g ∧
      MeasureTheory.Integrable (fun x => (g x).toReal) μ ∧
        (∀ᵐ (x : α) ∂μ, g x < ⊤) ∧ ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), f x ∂μ + ε
-- MeasureTheory.exists_upperSemicontinuous_lt_integral_gt
theorem exists_upperSemicontinuous_lt_integral_gt [SigmaFinite μ] (f : α → ℝ) (hf : Integrable f μ)
    {ε : ℝ} (εpos : 0 < ε) :
    ∃ g : α → EReal,
      (∀ x, (g x : EReal) < f x) ∧
      UpperSemicontinuous g ∧
      Integrable (fun x => EReal.toReal (g x)) μ ∧
      (∀ᵐ x ∂μ, ⊥ < g x) ∧ (∫ x, f x ∂μ) < (∫ x, EReal.toReal (g x) ∂μ) + ε 
Arguments: {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] {μ : MeasureTheory.Measure α} [μ.WeaklyRegular] [MeasureTheory.SigmaFinite μ] (f : α → ℝ) (hf : MeasureTheory.Integrable f μ) {ε : ℝ} (εpos : 0 < ε)
Goal: ∃ g,
  (∀ (x : α), g x < ↑(f x)) ∧
    UpperSemicontinuous g ∧
      MeasureTheory.Integrable (fun x => (g x).toReal) μ ∧
        (∀ᵐ (x : α) ∂μ, ⊥ < g x) ∧ ∫ (x : α), f x ∂μ < ∫ (x : α), (g x).toReal ∂μ + ε
-- MeasureTheory.countable_meas_le_ne_meas_lt
theorem countable_meas_le_ne_meas_lt (g : α → R) :
    {t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable 
Arguments: {α : Type u_1} {R : Type u_2} [MeasurableSpace α] (μ : MeasureTheory.Measure α) [LinearOrder R] (g : α → R)
Goal: {t | μ {a | t ≤ g a} ≠ μ {a | t < g a}}.Countable
-- MeasureTheory.meas_le_ae_eq_meas_lt
theorem meas_le_ae_eq_meas_lt {R : Type*} [LinearOrder R] [MeasurableSpace R]
    (ν : Measure R) [NoAtoms ν] (g : α → R) :
    (fun t => μ {a : α | t ≤ g a}) =ᵐ[ν] fun t => μ {a : α | t < g a} 
Arguments: {α : Type u_1} [MeasurableSpace α] (μ : MeasureTheory.Measure α) {R : Type u_3} [LinearOrder R] [MeasurableSpace R] (ν : MeasureTheory.Measure R) [MeasureTheory.NoAtoms ν] (g : α → R)
Goal: (fun t => μ {a | t ≤ g a}) =ᵐ[ν] fun t => μ {a | t < g a}
-- MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
    (μ : Measure α) [SFinite μ]
    (f_nn : 0 ≤ f) (f_mble : Measurable f)
    (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
    (g_nn : ∀ t > 0, 0 ≤ g t) :
    ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
      ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} {g : ℝ → ℝ} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (f_nn : 0 ≤ f) (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g MeasureTheory.volume 0 t) (g_mble : Measurable g) (g_nn : ∀ t > 0, 0 ≤ g t)
Goal: ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ =
  ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t)
-- MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable
theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable (μ : Measure α)
    (f_nn : 0 ≤ f) (f_mble : Measurable f)
    (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t) (g_mble : Measurable g)
    (g_nn : ∀ t > 0, 0 ≤ g t) :
    ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
      ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} {g : ℝ → ℝ} (μ : MeasureTheory.Measure α) (f_nn : 0 ≤ f) (f_mble : Measurable f) (g_intble : ∀ t > 0, IntervalIntegrable g MeasureTheory.volume 0 t) (g_mble : Measurable g) (g_nn : ∀ t > 0, 0 ≤ g t)
Goal: ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ =
  ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t)
-- MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul
theorem lintegral_comp_eq_lintegral_meas_le_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
    (f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
    (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
    ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
      ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (g t) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} {g : ℝ → ℝ} (μ : MeasureTheory.Measure α) (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g MeasureTheory.volume 0 t) (g_nn : ∀ᵐ (t : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioi 0), 0 ≤ g t)
Goal: ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ =
  ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t)
-- MeasureTheory.lintegral_eq_lintegral_meas_le
theorem lintegral_eq_lintegral_meas_le (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
    (f_mble : AEMeasurable f μ) :
    ∫⁻ ω, ENNReal.ofReal (f ω) ∂μ = ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} (μ : MeasureTheory.Measure α) (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ)
Goal: ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t ≤ f a}
-- MeasureTheory.lintegral_comp_eq_lintegral_meas_lt_mul
theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
    (f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
    (g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
    ∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
      ∫⁻ t in Ioi 0, μ {a : α | t < f a} * ENNReal.ofReal (g t) 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} {g : ℝ → ℝ} (μ : MeasureTheory.Measure α) (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g MeasureTheory.volume 0 t) (g_nn : ∀ᵐ (t : ℝ) ∂MeasureTheory.volume.restrict (Set.Ioi 0), 0 ≤ g t)
Goal: ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ =
  ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t < f a} * ENNReal.ofReal (g t)
-- MeasureTheory.lintegral_eq_lintegral_meas_lt
theorem lintegral_eq_lintegral_meas_lt (μ : Measure α)
    (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ) :
    ∫⁻ ω, ENNReal.ofReal (f ω) ∂μ = ∫⁻ t in Ioi 0, μ {a : α | t < f a} 
Arguments: {α : Type u_1} [MeasurableSpace α] {f : α → ℝ} (μ : MeasureTheory.Measure α) (f_nn : 0 ≤ᵐ[μ] f) (f_mble : AEMeasurable f μ)
Goal: ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Set.Ioi 0, μ {a | t < f a}
-- MeasureTheory.Integrable.integral_eq_integral_meas_lt
theorem Integrable.integral_eq_integral_meas_lt
    (f_intble : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
    ∫ ω, f ω ∂μ = ∫ t in Set.Ioi 0, ENNReal.toReal (μ {a : α | t < f a}) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f : α → ℝ} (f_intble : MeasureTheory.Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f)
Goal: ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Set.Ioi 0, (μ {a | t < f a}).toReal
-- MeasureTheory.Integrable.integral_eq_integral_meas_le
theorem Integrable.integral_eq_integral_meas_le
    (f_intble : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
    ∫ ω, f ω ∂μ = ∫ t in Set.Ioi 0, ENNReal.toReal (μ {a : α | t ≤ f a}) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f : α → ℝ} (f_intble : MeasureTheory.Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f)
Goal: ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Set.Ioi 0, (μ {a | t ≤ f a}).toReal
-- MeasureTheory.Integrable.integral_eq_integral_Ioc_meas_le
lemma Integrable.integral_eq_integral_Ioc_meas_le {f : α → ℝ} {M : ℝ}
    (f_intble : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) (f_bdd : f ≤ᵐ[μ] (fun _ ↦ M)) :
    ∫ ω, f ω ∂μ = ∫ t in Ioc 0 M, ENNReal.toReal (μ {a : α | t ≤ f a}) 
Arguments: {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f : α → ℝ} {M : ℝ} (f_intble : MeasureTheory.Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) (f_bdd : f ≤ᵐ[μ] fun x => M)
Goal: ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Set.Ioc 0 M, (μ {a | t ≤ f a}).toReal
-- DomMulAct.smul_aeeqFun_aeeq
@[to_additive]
theorem smul_aeeqFun_aeeq (c : Mᵈᵐᵃ) (f : α →ₘ[μ] β) :
    c • f =ᵐ[μ] (f <| mk.symm c • ·) 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace M] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SMul M α] [MeasurableSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] (c : Mᵈᵐᵃ) (f : α →ₘ[μ] β)
Goal: ↑(c • f) =ᵐ[μ] fun x => ↑f (DomMulAct.mk.symm c • x)
-- DomAddAct.vadd_aeeqFun_aeeq
@[to_additive]
theorem smul_aeeqFun_aeeq (c : Mᵈᵐᵃ) (f : α →ₘ[μ] β) :
    c • f =ᵐ[μ] (f <| mk.symm c • ·) 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace M] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [VAdd M α] [MeasurableVAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] (c : Mᵈᵃᵃ) (f : α →ₘ[μ] β)
Goal: ↑(c +ᵥ f) =ᵐ[μ] fun x => ↑f (DomAddAct.mk.symm c +ᵥ x)
-- DomMulAct.mk_smul_mk_aeeqFun
@[to_additive (attr := simp)]
theorem mk_smul_mk_aeeqFun (c : M) (f : α → β) (hf : AEStronglyMeasurable f μ) :
    mk c • AEEqFun.mk f hf = AEEqFun.mk (f <| c • ·)
      (hf.comp_measurePreserving (measurePreserving_smul _ _)) 
Arguments: {M : Type u_3} {α : Type u_1} {β : Type u_2} [MeasurableSpace M] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SMul M α] [MeasurableSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] (c : M) (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: DomMulAct.mk c • MeasureTheory.AEEqFun.mk f hf = MeasureTheory.AEEqFun.mk (fun x => f (c • x)) ⋯
-- DomAddAct.mk_vadd_mk_aeeqFun
@[to_additive (attr := simp)]
theorem mk_smul_mk_aeeqFun (c : M) (f : α → β) (hf : AEStronglyMeasurable f μ) :
    mk c • AEEqFun.mk f hf = AEEqFun.mk (f <| c • ·)
      (hf.comp_measurePreserving (measurePreserving_smul _ _)) 
Arguments: {M : Type u_3} {α : Type u_1} {β : Type u_2} [MeasurableSpace M] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [VAdd M α] [MeasurableVAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] (c : M) (f : α → β) (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: DomAddAct.mk c +ᵥ MeasureTheory.AEEqFun.mk f hf = MeasureTheory.AEEqFun.mk (fun x => f (c +ᵥ x)) ⋯
-- DomAddAct.vadd_aeeqFun_const
@[to_additive (attr := simp)]
theorem smul_aeeqFun_const (c : Mᵈᵐᵃ) (b : β) :
    c • (AEEqFun.const α b : α →ₘ[μ] β) = AEEqFun.const α b 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace M] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [VAdd M α] [MeasurableVAdd M α] [MeasureTheory.VAddInvariantMeasure M α μ] (c : Mᵈᵃᵃ) (b : β)
Goal: c +ᵥ MeasureTheory.AEEqFun.const α b = MeasureTheory.AEEqFun.const α b
-- DomMulAct.smul_aeeqFun_const
@[to_additive (attr := simp)]
theorem smul_aeeqFun_const (c : Mᵈᵐᵃ) (b : β) :
    c • (AEEqFun.const α b : α →ₘ[μ] β) = AEEqFun.const α b 
Arguments: {M : Type u_1} {α : Type u_2} {β : Type u_3} [MeasurableSpace M] [MeasurableSpace α] {μ : MeasureTheory.Measure α} [TopologicalSpace β] [SMul M α] [MeasurableSMul M α] [MeasureTheory.SMulInvariantMeasure M α μ] (c : Mᵈᵐᵃ) (b : β)
Goal: c • MeasureTheory.AEEqFun.const α b = MeasureTheory.AEEqFun.const α b
-- measurableSet_integrable
theorem measurableSet_integrable [SFinite ν] ⦃f : α → β → E⦄
    (hf : StronglyMeasurable (uncurry f)) : MeasurableSet {x | Integrable (f x) ν} 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SFinite ν] ⦃f : α → β → E⦄ (hf : MeasureTheory.StronglyMeasurable (Function.uncurry f))
Goal: MeasurableSet {x | MeasureTheory.Integrable (f x) ν}
-- MeasureTheory.StronglyMeasurable.integral_prod_right
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SFinite ν] ⦃f : α → β → E⦄
    (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂ν 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasureTheory.SFinite ν] ⦃f : α → β → E⦄ (hf : MeasureTheory.StronglyMeasurable (Function.uncurry f))
Goal: MeasureTheory.StronglyMeasurable fun x => ∫ (y : β), f x y ∂ν
-- MeasureTheory.StronglyMeasurable.integral_prod_right'
theorem MeasureTheory.StronglyMeasurable.integral_prod_right' [SFinite ν] ⦃f : α × β → E⦄
    (hf : StronglyMeasurable f) : StronglyMeasurable fun x => ∫ y, f (x, y) ∂ν 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasureTheory.SFinite ν] ⦃f : α × β → E⦄ (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable fun x => ∫ (y : β), f (x, y) ∂ν
-- MeasureTheory.StronglyMeasurable.integral_prod_left
theorem MeasureTheory.StronglyMeasurable.integral_prod_left [SFinite μ] ⦃f : α → β → E⦄
    (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun y => ∫ x, f x y ∂μ 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasureTheory.SFinite μ] ⦃f : α → β → E⦄ (hf : MeasureTheory.StronglyMeasurable (Function.uncurry f))
Goal: MeasureTheory.StronglyMeasurable fun y => ∫ (x : α), f x y ∂μ
-- MeasureTheory.StronglyMeasurable.integral_prod_left'
theorem MeasureTheory.StronglyMeasurable.integral_prod_left' [SFinite μ] ⦃f : α × β → E⦄
    (hf : StronglyMeasurable f) : StronglyMeasurable fun y => ∫ x, f (x, y) ∂μ 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasureTheory.SFinite μ] ⦃f : α × β → E⦄ (hf : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.StronglyMeasurable fun y => ∫ (x : α), f (x, y) ∂μ
-- MeasureTheory.Measure.integrable_measure_prod_mk_left
theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
    (h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] {s : Set (α × β)} (hs : MeasurableSet s) (h2s : (μ.prod ν) s ≠ ⊤)
Goal: MeasureTheory.Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ
-- MeasureTheory.AEStronglyMeasurable.prod_swap
nonrec theorem MeasureTheory.AEStronglyMeasurable.prod_swap {γ : Type*} [TopologicalSpace γ]
    [SFinite μ] [SFinite ν] {f : β × α → γ} (hf : AEStronglyMeasurable f (ν.prod μ)) :
    AEStronglyMeasurable (fun z : α × β => f z.swap) (μ.prod ν) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {γ : Type u_7} [TopologicalSpace γ] [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] {f : β × α → γ} (hf : MeasureTheory.AEStronglyMeasurable f (ν.prod μ))
Goal: MeasureTheory.AEStronglyMeasurable (fun z => f z.swap) (μ.prod ν)
-- MeasureTheory.AEStronglyMeasurable.fst
theorem MeasureTheory.AEStronglyMeasurable.fst {γ} [TopologicalSpace γ] [SFinite ν] {f : α → γ}
    (hf : AEStronglyMeasurable f μ) : AEStronglyMeasurable (fun z : α × β => f z.1) (μ.prod ν) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {γ : Type u_7} [TopologicalSpace γ] [MeasureTheory.SFinite ν] {f : α → γ} (hf : MeasureTheory.AEStronglyMeasurable f μ)
Goal: MeasureTheory.AEStronglyMeasurable (fun z => f z.1) (μ.prod ν)
-- MeasureTheory.AEStronglyMeasurable.snd
theorem MeasureTheory.AEStronglyMeasurable.snd {γ} [TopologicalSpace γ] [SFinite ν] {f : β → γ}
    (hf : AEStronglyMeasurable f ν) : AEStronglyMeasurable (fun z : α × β => f z.2) (μ.prod ν) 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {γ : Type u_7} [TopologicalSpace γ] [MeasureTheory.SFinite ν] {f : β → γ} (hf : MeasureTheory.AEStronglyMeasurable f ν)
Goal: MeasureTheory.AEStronglyMeasurable (fun z => f z.2) (μ.prod ν)
-- MeasureTheory.AEStronglyMeasurable.integral_prod_right'
theorem MeasureTheory.AEStronglyMeasurable.integral_prod_right' [SFinite ν] [NormedSpace ℝ E]
    ⦃f : α × β → E⦄ (hf : AEStronglyMeasurable f (μ.prod ν)) :
    AEStronglyMeasurable (fun x => ∫ y, f (x, y) ∂ν) μ 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SFinite ν] [NormedSpace ℝ E] ⦃f : α × β → E⦄ (hf : MeasureTheory.AEStronglyMeasurable f (μ.prod ν))
Goal: MeasureTheory.AEStronglyMeasurable (fun x => ∫ (y : β), f (x, y) ∂ν) μ
-- MeasureTheory.AEStronglyMeasurable.prod_mk_left
theorem MeasureTheory.AEStronglyMeasurable.prod_mk_left {γ : Type*} [SFinite ν]
    [TopologicalSpace γ] {f : α × β → γ} (hf : AEStronglyMeasurable f (μ.prod ν)) :
    ∀ᵐ x ∂μ, AEStronglyMeasurable (fun y => f (x, y)) ν 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {γ : Type u_7} [MeasureTheory.SFinite ν] [TopologicalSpace γ] {f : α × β → γ} (hf : MeasureTheory.AEStronglyMeasurable f (μ.prod ν))
Goal: ∀ᵐ (x : α) ∂μ, MeasureTheory.AEStronglyMeasurable (fun y => f (x, y)) ν
-- MeasureTheory.integrable_swap_iff
theorem integrable_swap_iff [SFinite μ] {f : α × β → E} :
    Integrable (f ∘ Prod.swap) (ν.prod μ) ↔ Integrable f (μ.prod ν) 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] {f : α × β → E}
Goal: MeasureTheory.Integrable (f ∘ Prod.swap) (ν.prod μ) ↔ MeasureTheory.Integrable f (μ.prod ν)
-- MeasureTheory.Integrable.swap
theorem Integrable.swap [SFinite μ] ⦃f : α × β → E⦄ (hf : Integrable f (μ.prod ν)) :
    Integrable (f ∘ Prod.swap) (ν.prod μ) 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] ⦃f : α × β → E⦄ (hf : MeasureTheory.Integrable f (μ.prod ν))
Goal: MeasureTheory.Integrable (f ∘ Prod.swap) (ν.prod μ)
-- MeasureTheory.hasFiniteIntegral_prod_iff
theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasurable f) :
    HasFiniteIntegral f (μ.prod ν) ↔
      (∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
        HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SFinite ν] ⦃f : α × β → E⦄ (h1f : MeasureTheory.StronglyMeasurable f)
Goal: MeasureTheory.HasFiniteIntegral f (μ.prod ν) ↔
  (∀ᵐ (x : α) ∂μ, MeasureTheory.HasFiniteIntegral (fun y => f (x, y)) ν) ∧
    MeasureTheory.HasFiniteIntegral (fun x => ∫ (y : β), ‖f (x, y)‖ ∂ν) μ
-- MeasureTheory.hasFiniteIntegral_prod_iff'
theorem hasFiniteIntegral_prod_iff' ⦃f : α × β → E⦄ (h1f : AEStronglyMeasurable f (μ.prod ν)) :
    HasFiniteIntegral f (μ.prod ν) ↔
      (∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
        HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SFinite ν] ⦃f : α × β → E⦄ (h1f : MeasureTheory.AEStronglyMeasurable f (μ.prod ν))
Goal: MeasureTheory.HasFiniteIntegral f (μ.prod ν) ↔
  (∀ᵐ (x : α) ∂μ, MeasureTheory.HasFiniteIntegral (fun y => f (x, y)) ν) ∧
    MeasureTheory.HasFiniteIntegral (fun x => ∫ (y : β), ‖f (x, y)‖ ∂ν) μ
-- MeasureTheory.integrable_prod_iff
theorem integrable_prod_iff ⦃f : α × β → E⦄ (h1f : AEStronglyMeasurable f (μ.prod ν)) :
    Integrable f (μ.prod ν) ↔
      (∀ᵐ x ∂μ, Integrable (fun y => f (x, y)) ν) ∧ Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SFinite ν] ⦃f : α × β → E⦄ (h1f : MeasureTheory.AEStronglyMeasurable f (μ.prod ν))
Goal: MeasureTheory.Integrable f (μ.prod ν) ↔
  (∀ᵐ (x : α) ∂μ, MeasureTheory.Integrable (fun y => f (x, y)) ν) ∧
    MeasureTheory.Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂ν) μ
-- MeasureTheory.integrable_prod_iff'
theorem integrable_prod_iff' [SFinite μ] ⦃f : α × β → E⦄
    (h1f : AEStronglyMeasurable f (μ.prod ν)) :
    Integrable f (μ.prod ν) ↔
      (∀ᵐ y ∂ν, Integrable (fun x => f (x, y)) μ) ∧ Integrable (fun y => ∫ x, ‖f (x, y)‖ ∂μ) ν 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] ⦃f : α × β → E⦄ (h1f : MeasureTheory.AEStronglyMeasurable f (μ.prod ν))
Goal: MeasureTheory.Integrable f (μ.prod ν) ↔
  (∀ᵐ (y : β) ∂ν, MeasureTheory.Integrable (fun x => f (x, y)) μ) ∧
    MeasureTheory.Integrable (fun y => ∫ (x : α), ‖f (x, y)‖ ∂μ) ν
-- MeasureTheory.Integrable.prod_left_ae
theorem Integrable.prod_left_ae [SFinite μ] ⦃f : α × β → E⦄ (hf : Integrable f (μ.prod ν)) :
    ∀ᵐ y ∂ν, Integrable (fun x => f (x, y)) μ 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] ⦃f : α × β → E⦄ (hf : MeasureTheory.Integrable f (μ.prod ν))
Goal: ∀ᵐ (y : β) ∂ν, MeasureTheory.Integrable (fun x => f (x, y)) μ
-- MeasureTheory.Integrable.prod_right_ae
theorem Integrable.prod_right_ae [SFinite μ] ⦃f : α × β → E⦄ (hf : Integrable f (μ.prod ν)) :
    ∀ᵐ x ∂μ, Integrable (fun y => f (x, y)) ν 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] ⦃f : α × β → E⦄ (hf : MeasureTheory.Integrable f (μ.prod ν))
Goal: ∀ᵐ (x : α) ∂μ, MeasureTheory.Integrable (fun y => f (x, y)) ν
-- MeasureTheory.Integrable.integral_norm_prod_left
theorem Integrable.integral_norm_prod_left ⦃f : α × β → E⦄ (hf : Integrable f (μ.prod ν)) :
    Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SFinite ν] ⦃f : α × β → E⦄ (hf : MeasureTheory.Integrable f (μ.prod ν))
Goal: MeasureTheory.Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂ν) μ
-- MeasureTheory.Integrable.integral_norm_prod_right
theorem Integrable.integral_norm_prod_right [SFinite μ] ⦃f : α × β → E⦄
    (hf : Integrable f (μ.prod ν)) : Integrable (fun y => ∫ x, ‖f (x, y)‖ ∂μ) ν 
Arguments: {α : Type u_1} {β : Type u_3} {E : Type u_6} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] ⦃f : α × β → E⦄ (hf : MeasureTheory.Integrable f (μ.prod ν))
Goal: MeasureTheory.Integrable (fun y => ∫ (x : α), ‖f (x, y)‖ ∂μ) ν