ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Integrable.prod_smul
theorem Integrable.prod_smul {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E]
    {f : α → 𝕜} {g : β → E} (hf : Integrable f μ) (hg : Integrable g ν) :
    Integrable (fun z : α × β => f z.1 • g z.2) (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] {𝕜 : Type u_7} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] {f : α → 𝕜} {g : β → E} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g ν)
Goal: MeasureTheory.Integrable (fun z => f z.1 • g z.2) (μ.prod ν)
-- MeasureTheory.Integrable.prod_mul
theorem Integrable.prod_mul {L : Type*} [RCLike L] {f : α → L} {g : β → L} (hf : Integrable f μ)
    (hg : Integrable g ν) : Integrable (fun z : α × β => f z.1 * g 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} [MeasureTheory.SFinite ν] {L : Type u_7} [RCLike L] {f : α → L} {g : β → L} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g ν)
Goal: MeasureTheory.Integrable (fun z => f z.1 * g z.2) (μ.prod ν)
-- MeasureTheory.Integrable.integral_prod_left
theorem Integrable.integral_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] [NormedSpace ℝ E] ⦃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_prod_right
theorem Integrable.integral_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] [NormedSpace ℝ E] [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
-- MeasureTheory.integral_prod_swap
theorem integral_prod_swap (f : α × β → E) :
    ∫ z, f z.swap ∂ν.prod μ = ∫ z, f z ∂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] [NormedSpace ℝ E] [MeasureTheory.SFinite μ] (f : α × β → E)
Goal: ∫ (z : β × α), f z.swap ∂ν.prod μ = ∫ (z : α × β), f z ∂μ.prod ν
-- MeasureTheory.integral_fn_integral_add
theorem integral_fn_integral_add ⦃f g : α × β → E⦄ (F : E → E') (hf : Integrable f (μ.prod ν))
    (hg : Integrable g (μ.prod ν)) :
    (∫ x, F (∫ y, f (x, y) + g (x, y) ∂ν) ∂μ) =
      ∫ x, F ((∫ y, f (x, y) ∂ν) + ∫ y, g (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] [MeasureTheory.SFinite μ] {E' : Type u_7} [NormedAddCommGroup E'] [NormedSpace ℝ E'] ⦃f g : α × β → E⦄ (F : E → E') (hf : MeasureTheory.Integrable f (μ.prod ν)) (hg : MeasureTheory.Integrable g (μ.prod ν))
Goal: ∫ (x : α), F (∫ (y : β), f (x, y) + g (x, y) ∂ν) ∂μ = ∫ (x : α), F (∫ (y : β), f (x, y) ∂ν + ∫ (y : β), g (x, y) ∂ν) ∂μ
-- MeasureTheory.integral_fn_integral_sub
theorem integral_fn_integral_sub ⦃f g : α × β → E⦄ (F : E → E') (hf : Integrable f (μ.prod ν))
    (hg : Integrable g (μ.prod ν)) :
    (∫ x, F (∫ y, f (x, y) - g (x, y) ∂ν) ∂μ) =
      ∫ x, F ((∫ y, f (x, y) ∂ν) - ∫ y, g (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] [MeasureTheory.SFinite μ] {E' : Type u_7} [NormedAddCommGroup E'] [NormedSpace ℝ E'] ⦃f g : α × β → E⦄ (F : E → E') (hf : MeasureTheory.Integrable f (μ.prod ν)) (hg : MeasureTheory.Integrable g (μ.prod ν))
Goal: ∫ (x : α), F (∫ (y : β), f (x, y) - g (x, y) ∂ν) ∂μ = ∫ (x : α), F (∫ (y : β), f (x, y) ∂ν - ∫ (y : β), g (x, y) ∂ν) ∂μ
-- MeasureTheory.lintegral_fn_integral_sub
theorem lintegral_fn_integral_sub ⦃f g : α × β → E⦄ (F : E → ℝ≥0∞) (hf : Integrable f (μ.prod ν))
    (hg : Integrable g (μ.prod ν)) :
    (∫⁻ x, F (∫ y, f (x, y) - g (x, y) ∂ν) ∂μ) =
      ∫⁻ x, F ((∫ y, f (x, y) ∂ν) - ∫ y, g (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] [MeasureTheory.SFinite μ] ⦃f g : α × β → E⦄ (F : E → ℝ≥0∞) (hf : MeasureTheory.Integrable f (μ.prod ν)) (hg : MeasureTheory.Integrable g (μ.prod ν))
Goal: ∫⁻ (x : α), F (∫ (y : β), f (x, y) - g (x, y) ∂ν) ∂μ =
  ∫⁻ (x : α), F (∫ (y : β), f (x, y) ∂ν - ∫ (y : β), g (x, y) ∂ν) ∂μ
-- MeasureTheory.integral_integral_add
theorem integral_integral_add ⦃f g : α × β → E⦄ (hf : Integrable f (μ.prod ν))
    (hg : Integrable g (μ.prod ν)) :
    (∫ x, ∫ y, f (x, y) + g (x, y) ∂ν ∂μ) = (∫ x, ∫ y, f (x, y) ∂ν ∂μ) + ∫ x, ∫ y, g (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] [MeasureTheory.SFinite μ] ⦃f g : α × β → E⦄ (hf : MeasureTheory.Integrable f (μ.prod ν)) (hg : MeasureTheory.Integrable g (μ.prod ν))
Goal: ∫ (x : α), ∫ (y : β), f (x, y) + g (x, y) ∂ν ∂μ =
  ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ + ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ
-- MeasureTheory.integral_integral_add'
theorem integral_integral_add' ⦃f g : α × β → E⦄ (hf : Integrable f (μ.prod ν))
    (hg : Integrable g (μ.prod ν)) :
    (∫ x, ∫ y, (f + g) (x, y) ∂ν ∂μ) = (∫ x, ∫ y, f (x, y) ∂ν ∂μ) + ∫ x, ∫ y, g (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] [MeasureTheory.SFinite μ] ⦃f g : α × β → E⦄ (hf : MeasureTheory.Integrable f (μ.prod ν)) (hg : MeasureTheory.Integrable g (μ.prod ν))
Goal: ∫ (x : α), ∫ (y : β), (f + g) (x, y) ∂ν ∂μ = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ + ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ
-- MeasureTheory.integral_integral_sub
theorem integral_integral_sub ⦃f g : α × β → E⦄ (hf : Integrable f (μ.prod ν))
    (hg : Integrable g (μ.prod ν)) :
    (∫ x, ∫ y, f (x, y) - g (x, y) ∂ν ∂μ) = (∫ x, ∫ y, f (x, y) ∂ν ∂μ) - ∫ x, ∫ y, g (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] [MeasureTheory.SFinite μ] ⦃f g : α × β → E⦄ (hf : MeasureTheory.Integrable f (μ.prod ν)) (hg : MeasureTheory.Integrable g (μ.prod ν))
Goal: ∫ (x : α), ∫ (y : β), f (x, y) - g (x, y) ∂ν ∂μ =
  ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ - ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ
-- MeasureTheory.integral_integral_sub'
theorem integral_integral_sub' ⦃f g : α × β → E⦄ (hf : Integrable f (μ.prod ν))
    (hg : Integrable g (μ.prod ν)) :
    (∫ x, ∫ y, (f - g) (x, y) ∂ν ∂μ) = (∫ x, ∫ y, f (x, y) ∂ν ∂μ) - ∫ x, ∫ y, g (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] [MeasureTheory.SFinite μ] ⦃f g : α × β → E⦄ (hf : MeasureTheory.Integrable f (μ.prod ν)) (hg : MeasureTheory.Integrable g (μ.prod ν))
Goal: ∫ (x : α), ∫ (y : β), (f - g) (x, y) ∂ν ∂μ = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ - ∫ (x : α), ∫ (y : β), g (x, y) ∂ν ∂μ
-- MeasureTheory.continuous_integral_integral
theorem continuous_integral_integral :
    Continuous fun f : α × β →₁[μ.prod ν] E => ∫ 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] [MeasureTheory.SFinite μ]
Goal: Continuous fun f => ∫ (x : α), ∫ (y : β), ↑↑f (x, y) ∂ν ∂μ
-- MeasureTheory.integral_prod
theorem integral_prod (f : α × β → E) (hf : Integrable f (μ.prod ν)) :
    ∫ z, f z ∂μ.prod ν = ∫ 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] [MeasureTheory.SFinite μ] (f : α × β → E) (hf : MeasureTheory.Integrable f (μ.prod ν))
Goal: ∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ
-- MeasureTheory.integral_prod_symm
theorem integral_prod_symm (f : α × β → E) (hf : Integrable f (μ.prod ν)) :
    ∫ z, f z ∂μ.prod ν = ∫ 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] [NormedSpace ℝ E] [MeasureTheory.SFinite μ] (f : α × β → E) (hf : MeasureTheory.Integrable f (μ.prod ν))
Goal: ∫ (z : α × β), f z ∂μ.prod ν = ∫ (y : β), ∫ (x : α), f (x, y) ∂μ ∂ν
-- MeasureTheory.integral_integral
theorem integral_integral {f : α → β → E} (hf : Integrable (uncurry f) (μ.prod ν)) :
    ∫ x, ∫ y, f x y ∂ν ∂μ = ∫ z, f z.1 z.2 ∂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] [NormedSpace ℝ E] [MeasureTheory.SFinite μ] {f : α → β → E} (hf : MeasureTheory.Integrable (Function.uncurry f) (μ.prod ν))
Goal: ∫ (x : α), ∫ (y : β), f x y ∂ν ∂μ = ∫ (z : α × β), f z.1 z.2 ∂μ.prod ν
-- MeasureTheory.integral_integral_symm
theorem integral_integral_symm {f : α → β → E} (hf : Integrable (uncurry f) (μ.prod ν)) :
    ∫ x, ∫ y, f x y ∂ν ∂μ = ∫ z, f z.2 z.1 ∂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] [NormedSpace ℝ E] [MeasureTheory.SFinite μ] {f : α → β → E} (hf : MeasureTheory.Integrable (Function.uncurry f) (μ.prod ν))
Goal: ∫ (x : α), ∫ (y : β), f x y ∂ν ∂μ = ∫ (z : β × α), f z.2 z.1 ∂ν.prod μ
-- MeasureTheory.integral_integral_swap
theorem integral_integral_swap ⦃f : α → β → E⦄ (hf : Integrable (uncurry f) (μ.prod ν)) :
    ∫ x, ∫ y, f x y ∂ν ∂μ = ∫ 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] [NormedSpace ℝ E] [MeasureTheory.SFinite μ] ⦃f : α → β → E⦄ (hf : MeasureTheory.Integrable (Function.uncurry f) (μ.prod ν))
Goal: ∫ (x : α), ∫ (y : β), f x y ∂ν ∂μ = ∫ (y : β), ∫ (x : α), f x y ∂μ ∂ν
-- MeasureTheory.setIntegral_prod
theorem setIntegral_prod (f : α × β → E) {s : Set α} {t : Set β}
    (hf : IntegrableOn f (s ×ˢ t) (μ.prod ν)) :
    ∫ z in s ×ˢ t, f z ∂μ.prod ν = ∫ x in s, ∫ y in t, 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] [MeasureTheory.SFinite μ] (f : α × β → E) {s : Set α} {t : Set β} (hf : MeasureTheory.IntegrableOn f (s ×ˢ t) (μ.prod ν))
Goal: ∫ (z : α × β) in s ×ˢ t, f z ∂μ.prod ν = ∫ (x : α) in s, ∫ (y : β) in t, f (x, y) ∂ν ∂μ
-- MeasureTheory.integral_prod_smul
theorem integral_prod_smul {𝕜 : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] (f : α → 𝕜) (g : β → E) :
    ∫ z, f z.1 • g z.2 ∂μ.prod ν = (∫ x, f x ∂μ) • ∫ y, g 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] [NormedSpace ℝ E] [MeasureTheory.SFinite μ] {𝕜 : Type u_8} [RCLike 𝕜] [NormedSpace 𝕜 E] (f : α → 𝕜) (g : β → E)
Goal: ∫ (z : α × β), f z.1 • g z.2 ∂μ.prod ν = (∫ (x : α), f x ∂μ) • ∫ (y : β), g y ∂ν
-- MeasureTheory.integral_prod_mul
theorem integral_prod_mul {L : Type*} [RCLike L] (f : α → L) (g : β → L) :
    ∫ z, f z.1 * g z.2 ∂μ.prod ν = (∫ x, f x ∂μ) * ∫ y, g y ∂ν 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] {L : Type u_8} [RCLike L] (f : α → L) (g : β → L)
Goal: ∫ (z : α × β), f z.1 * g z.2 ∂μ.prod ν = (∫ (x : α), f x ∂μ) * ∫ (y : β), g y ∂ν
-- MeasureTheory.setIntegral_prod_mul
theorem setIntegral_prod_mul {L : Type*} [RCLike L] (f : α → L) (g : β → L) (s : Set α)
    (t : Set β) :
    ∫ z in s ×ˢ t, f z.1 * g z.2 ∂μ.prod ν = (∫ x in s, f x ∂μ) * ∫ y in t, g y ∂ν 
Arguments: {α : Type u_1} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SFinite ν] [MeasureTheory.SFinite μ] {L : Type u_8} [RCLike L] (f : α → L) (g : β → L) (s : Set α) (t : Set β)
Goal: ∫ (z : α × β) in s ×ˢ t, f z.1 * g z.2 ∂μ.prod ν = (∫ (x : α) in s, f x ∂μ) * ∫ (y : β) in t, g y ∂ν
-- MeasureTheory.integral_fun_snd
theorem integral_fun_snd (f : β → E) : ∫ z, f z.2 ∂μ.prod ν = (μ univ).toReal • ∫ y, f 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] [NormedSpace ℝ E] [MeasureTheory.SFinite μ] (f : β → E)
Goal: ∫ (z : α × β), f z.2 ∂μ.prod ν = (μ Set.univ).toReal • ∫ (y : β), f y ∂ν
-- MeasureTheory.integral_fun_fst
theorem integral_fun_fst (f : α → E) : ∫ z, f z.1 ∂μ.prod ν = (ν univ).toReal • ∫ x, f x ∂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] [MeasureTheory.SFinite μ] (f : α → E)
Goal: ∫ (z : α × β), f z.1 ∂μ.prod ν = (ν Set.univ).toReal • ∫ (x : α), f x ∂μ
-- MeasureTheory.integral_integral_swap_of_hasCompactSupport
lemma integral_integral_swap_of_hasCompactSupport
    {f : X → Y → E} (hf : Continuous f.uncurry) (h'f : HasCompactSupport f.uncurry)
    {μ : Measure X} {ν : Measure Y} [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν] :
    ∫ x, (∫ y, f x y ∂ν) ∂μ = ∫ y, (∫ x, f x y ∂μ) ∂ν 
Arguments: {E : Type u_6} [NormedAddCommGroup E] [NormedSpace ℝ E] {X : Type u_8} {Y : Type u_9} [TopologicalSpace X] [TopologicalSpace Y] [MeasurableSpace X] [MeasurableSpace Y] [OpensMeasurableSpace X] [OpensMeasurableSpace Y] {f : X → Y → E} (hf : Continuous (Function.uncurry f)) (h'f : HasCompactSupport (Function.uncurry f)) {μ : MeasureTheory.Measure X} {ν : MeasureTheory.Measure Y} [MeasureTheory.IsFiniteMeasureOnCompacts μ] [MeasureTheory.IsFiniteMeasureOnCompacts ν]
Goal: ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν
-- intervalIntegrable_iff
theorem intervalIntegrable_iff : IntervalIntegrable f μ a b ↔ IntegrableOn f (Ι a b) μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ}
Goal: IntervalIntegrable f μ a b ↔ MeasureTheory.IntegrableOn f (Ι a b) μ
-- IntervalIntegrable.def'
theorem IntervalIntegrable.def' (h : IntervalIntegrable f μ a b) : IntegrableOn f (Ι a b) μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (h : IntervalIntegrable f μ a b)
Goal: MeasureTheory.IntegrableOn f (Ι a b) μ
-- intervalIntegrable_iff_integrableOn_Ioc_of_le
theorem intervalIntegrable_iff_integrableOn_Ioc_of_le (hab : a ≤ b) :
    IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioc a b) μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hab : a ≤ b)
Goal: IntervalIntegrable f μ a b ↔ MeasureTheory.IntegrableOn f (Set.Ioc a b) μ
-- intervalIntegrable_iff'
theorem intervalIntegrable_iff' [NoAtoms μ] :
    IntervalIntegrable f μ a b ↔ IntegrableOn f (uIcc a b) μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.NoAtoms μ]
Goal: IntervalIntegrable f μ a b ↔ MeasureTheory.IntegrableOn f [[a, b]] μ
-- intervalIntegrable_iff_integrableOn_Icc_of_le
theorem intervalIntegrable_iff_integrableOn_Icc_of_le {f : ℝ → E} {a b : ℝ} (hab : a ≤ b)
    {μ : Measure ℝ} [NoAtoms μ] : IntervalIntegrable f μ a b ↔ IntegrableOn f (Icc a b) μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} (hab : a ≤ b) {μ : MeasureTheory.Measure ℝ} [MeasureTheory.NoAtoms μ]
Goal: IntervalIntegrable f μ a b ↔ MeasureTheory.IntegrableOn f (Set.Icc a b) μ
-- intervalIntegrable_iff_integrableOn_Ico_of_le
theorem intervalIntegrable_iff_integrableOn_Ico_of_le [NoAtoms μ] (hab : a ≤ b) :
    IntervalIntegrable f μ a b ↔ IntegrableOn f (Ico a b) μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.NoAtoms μ] (hab : a ≤ b)
Goal: IntervalIntegrable f μ a b ↔ MeasureTheory.IntegrableOn f (Set.Ico a b) μ
-- intervalIntegrable_iff_integrableOn_Ioo_of_le
theorem intervalIntegrable_iff_integrableOn_Ioo_of_le [NoAtoms μ] (hab : a ≤ b) :
    IntervalIntegrable f μ a b ↔ IntegrableOn f (Ioo a b) μ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.NoAtoms μ] (hab : a ≤ b)
Goal: IntervalIntegrable f μ a b ↔ MeasureTheory.IntegrableOn f (Set.Ioo a b) μ
-- MeasureTheory.Integrable.intervalIntegrable
theorem MeasureTheory.Integrable.intervalIntegrable (hf : Integrable f μ) :
    IntervalIntegrable f μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hf : MeasureTheory.Integrable f μ)
Goal: IntervalIntegrable f μ a b
-- MeasureTheory.IntegrableOn.intervalIntegrable
theorem MeasureTheory.IntegrableOn.intervalIntegrable (hf : IntegrableOn f [[a, b]] μ) :
    IntervalIntegrable f μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (hf : MeasureTheory.IntegrableOn f [[a, b]] μ)
Goal: IntervalIntegrable f μ a b
-- intervalIntegrable_const_iff
theorem intervalIntegrable_const_iff {c : E} :
    IntervalIntegrable (fun _ => c) μ a b ↔ c = 0 ∨ μ (Ι a b) < ∞ 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {c : E}
Goal: IntervalIntegrable (fun x => c) μ a b ↔ c = 0 ∨ μ (Ι a b) < ⊤
-- intervalIntegrable_const
@[simp] -- Porting note (#10618): simp can prove this
theorem intervalIntegrable_const [IsLocallyFiniteMeasure μ] {c : E} :
    IntervalIntegrable (fun _ => c) μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {a b : ℝ} {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsLocallyFiniteMeasure μ] {c : E}
Goal: IntervalIntegrable (fun x => c) μ a b
-- IntervalIntegrable.symm
@[symm]
nonrec theorem symm (h : IntervalIntegrable f μ a b) : IntervalIntegrable f μ b a 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (h : IntervalIntegrable f μ a b)
Goal: IntervalIntegrable f μ b a
-- IntervalIntegrable.refl
@[refl, simp] -- Porting note: added `simp`
theorem refl : IntervalIntegrable f μ a a 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a : ℝ} {μ : MeasureTheory.Measure ℝ}
Goal: IntervalIntegrable f μ a a
-- IntervalIntegrable.trans
@[trans]
theorem trans {a b c : ℝ} (hab : IntervalIntegrable f μ a b) (hbc : IntervalIntegrable f μ b c) :
    IntervalIntegrable f μ a c 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {a b c : ℝ} (hab : IntervalIntegrable f μ a b) (hbc : IntervalIntegrable f μ b c)
Goal: IntervalIntegrable f μ a c
-- IntervalIntegrable.trans_iterate_Ico
theorem trans_iterate_Ico {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n)
    (hint : ∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a <| k + 1)) :
    IntervalIntegrable f μ (a m) (a n) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {a : ℕ → ℝ} {m n : ℕ} (hmn : m ≤ n) (hint : ∀ k ∈ Set.Ico m n, IntervalIntegrable f μ (a k) (a (k + 1)))
Goal: IntervalIntegrable f μ (a m) (a n)
-- IntervalIntegrable.trans_iterate
theorem trans_iterate {a : ℕ → ℝ} {n : ℕ}
    (hint : ∀ k < n, IntervalIntegrable f μ (a k) (a <| k + 1)) :
    IntervalIntegrable f μ (a 0) (a n) 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {a : ℕ → ℝ} {n : ℕ} (hint : ∀ k < n, IntervalIntegrable f μ (a k) (a (k + 1)))
Goal: IntervalIntegrable f μ (a 0) (a n)
-- IntervalIntegrable.neg
theorem neg (h : IntervalIntegrable f μ a b) : IntervalIntegrable (-f) μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (h : IntervalIntegrable f μ a b)
Goal: IntervalIntegrable (-f) μ a b
-- IntervalIntegrable.norm
theorem norm (h : IntervalIntegrable f μ a b) : IntervalIntegrable (fun x => ‖f x‖) μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ : MeasureTheory.Measure ℝ} (h : IntervalIntegrable f μ a b)
Goal: IntervalIntegrable (fun x => ‖f x‖) μ a b
-- IntervalIntegrable.intervalIntegrable_norm_iff
theorem intervalIntegrable_norm_iff {f : ℝ → E} {μ : Measure ℝ} {a b : ℝ}
    (hf : AEStronglyMeasurable f (μ.restrict (Ι a b))) :
    IntervalIntegrable (fun t => ‖f t‖) μ a b ↔ IntervalIntegrable f μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {μ : MeasureTheory.Measure ℝ} {a b : ℝ} (hf : MeasureTheory.AEStronglyMeasurable f (μ.restrict (Ι a b)))
Goal: IntervalIntegrable (fun t => ‖f t‖) μ a b ↔ IntervalIntegrable f μ a b
-- IntervalIntegrable.abs
theorem abs {f : ℝ → ℝ} (h : IntervalIntegrable f μ a b) :
    IntervalIntegrable (fun x => |f x|) μ a b 
Arguments: {a b : ℝ} {μ : MeasureTheory.Measure ℝ} {f : ℝ → ℝ} (h : IntervalIntegrable f μ a b)
Goal: IntervalIntegrable (fun x => |f x|) μ a b
-- IntervalIntegrable.mono
theorem mono (hf : IntervalIntegrable f ν a b) (h1 : [[c, d]] ⊆ [[a, b]]) (h2 : μ ≤ ν) :
    IntervalIntegrable f μ c d 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b c d : ℝ} {μ ν : MeasureTheory.Measure ℝ} (hf : IntervalIntegrable f ν a b) (h1 : [[c, d]] ⊆ [[a, b]]) (h2 : μ ≤ ν)
Goal: IntervalIntegrable f μ c d
-- IntervalIntegrable.mono_measure
theorem mono_measure (hf : IntervalIntegrable f ν a b) (h : μ ≤ ν) : IntervalIntegrable f μ a b 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b : ℝ} {μ ν : MeasureTheory.Measure ℝ} (hf : IntervalIntegrable f ν a b) (h : μ ≤ ν)
Goal: IntervalIntegrable f μ a b
-- IntervalIntegrable.mono_set
theorem mono_set (hf : IntervalIntegrable f μ a b) (h : [[c, d]] ⊆ [[a, b]]) :
    IntervalIntegrable f μ c d 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b c d : ℝ} {μ : MeasureTheory.Measure ℝ} (hf : IntervalIntegrable f μ a b) (h : [[c, d]] ⊆ [[a, b]])
Goal: IntervalIntegrable f μ c d
-- IntervalIntegrable.mono_set_ae
theorem mono_set_ae (hf : IntervalIntegrable f μ a b) (h : Ι c d ≤ᵐ[μ] Ι a b) :
    IntervalIntegrable f μ c d 
Arguments: {E : Type u_3} [NormedAddCommGroup E] {f : ℝ → E} {a b c d : ℝ} {μ : MeasureTheory.Measure ℝ} (hf : IntervalIntegrable f μ a b) (h : Ι c d ≤ᵐ[μ] Ι a b)
Goal: IntervalIntegrable f μ c d