File: Mathlib/MeasureTheory/Constructions/Prod/Integral.lean

full_name: measurableSet_integrable
code: theorem measurableSet_integrable [SFinite ν] ⦃f : α → β → E⦄
    (hf : StronglyMeasurable (uncurry f)) : MeasurableSet {x | Integrable (f x) ν} 
goal: MeasurableSet {x | MeasureTheory.Integrable (f x) ν}

full_name: MeasureTheory.StronglyMeasurable.integral_prod_right
code: /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
  Fubini's theorem is measurable.
  This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SFinite ν] ⦃f : α → β → E⦄
    (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂ν 
goal: MeasureTheory.StronglyMeasurable fun x => ∫ (y : β), f x y ∂ν

full_name: MeasureTheory.StronglyMeasurable.integral_prod_right'
code: /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
  Fubini's theorem is measurable. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right' [SFinite ν] ⦃f : α × β → E⦄
    (hf : StronglyMeasurable f) : StronglyMeasurable fun x => ∫ y, f (x, y) ∂ν 
goal: MeasureTheory.StronglyMeasurable fun x => ∫ (y : β), f (x, y) ∂ν

full_name: MeasureTheory.StronglyMeasurable.integral_prod_left
code: /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
  the symmetric version of Fubini's theorem is measurable.
  This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_left [SFinite μ] ⦃f : α → β → E⦄
    (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun y => ∫ x, f x y ∂μ 
goal: MeasureTheory.StronglyMeasurable fun y => ∫ (x : α), f x y ∂μ

full_name: MeasureTheory.StronglyMeasurable.integral_prod_left'
code: /-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
  the symmetric version of Fubini's theorem is measurable. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_left' [SFinite μ] ⦃f : α × β → E⦄
    (hf : StronglyMeasurable f) : StronglyMeasurable fun y => ∫ x, f (x, y) ∂μ 
goal: MeasureTheory.StronglyMeasurable fun y => ∫ (x : α), f (x, y) ∂μ

full_name: MeasureTheory.Measure.integrable_measure_prod_mk_left
code: 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 
goal: MeasureTheory.Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ

full_name: MeasureTheory.AEStronglyMeasurable.prod_swap
code: 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) 
goal: MeasureTheory.AEStronglyMeasurable (fun z => f z.swap) (μ.prod ν)

full_name: MeasureTheory.AEStronglyMeasurable.fst
code: 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) 
goal: MeasureTheory.AEStronglyMeasurable (fun z => f z.1) (μ.prod ν)

full_name: MeasureTheory.AEStronglyMeasurable.snd
code: 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) 
goal: MeasureTheory.AEStronglyMeasurable (fun z => f z.2) (μ.prod ν)

full_name: MeasureTheory.AEStronglyMeasurable.integral_prod_right'
code: /-- The Bochner integral is a.e.-measurable.
  This shows that the integrand of (the right-hand-side of) Fubini's theorem is a.e.-measurable. -/
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 
goal: MeasureTheory.AEStronglyMeasurable (fun x => ∫ (y : β), f (x, y) ∂ν) μ

full_name: MeasureTheory.AEStronglyMeasurable.prod_mk_left
code: 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 
goal: ∀ᵐ (x : α) ∂μ, MeasureTheory.AEStronglyMeasurable (fun y => f (x, y)) ν

full_name: MeasureTheory.integrable_swap_iff
code: theorem integrable_swap_iff [SFinite μ] {f : α × β → E} :
    Integrable (f ∘ Prod.swap) (ν.prod μ) ↔ Integrable f (μ.prod ν) 
goal: MeasureTheory.Integrable (f ∘ Prod.swap) (ν.prod μ) ↔ MeasureTheory.Integrable f (μ.prod ν)

full_name: MeasureTheory.Integrable.swap
code: theorem Integrable.swap [SFinite μ] ⦃f : α × β → E⦄ (hf : Integrable f (μ.prod ν)) :
    Integrable (f ∘ Prod.swap) (ν.prod μ) 
goal: MeasureTheory.Integrable (f ∘ Prod.swap) (ν.prod μ)

full_name: MeasureTheory.hasFiniteIntegral_prod_iff
code: 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 
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

full_name: MeasureTheory.hasFiniteIntegral_prod_iff'
code: 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 
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

full_name: MeasureTheory.integrable_prod_iff
code: /-- A binary function is integrable if the function `y ↦ f (x, y)` is integrable for almost every
  `x` and the function `x ↦ ∫ ‖f (x, y)‖ dy` is integrable. -/
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 
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

full_name: MeasureTheory.integrable_prod_iff'
code: /-- A binary function is integrable if the function `x ↦ f (x, y)` is integrable for almost every
  `y` and the function `y ↦ ∫ ‖f (x, y)‖ dx` is integrable. -/
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 
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

full_name: MeasureTheory.Integrable.prod_left_ae
code: 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 
goal: ∀ᵐ (y : β) ∂ν, MeasureTheory.Integrable (fun x => f (x, y)) μ

full_name: MeasureTheory.Integrable.prod_right_ae
code: 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 
goal: ∀ᵐ (x : α) ∂μ, MeasureTheory.Integrable (fun y => f (x, y)) ν

full_name: MeasureTheory.Integrable.integral_norm_prod_left
code: 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 
goal: MeasureTheory.Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂ν) μ

full_name: MeasureTheory.Integrable.integral_norm_prod_right
code: 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 
goal: MeasureTheory.Integrable (fun y => ∫ (x : α), ‖f (x, y)‖ ∂μ) ν

full_name: MeasureTheory.Integrable.prod_smul
code: 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) 
goal: MeasureTheory.Integrable (fun z => f z.1 • g z.2) (μ.prod ν)

full_name: MeasureTheory.Integrable.prod_mul
code: 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) 
goal: MeasureTheory.Integrable (fun z => f z.1 * g z.2) (μ.prod ν)

full_name: MeasureTheory.Integrable.integral_prod_left
code: theorem Integrable.integral_prod_left ⦃f : α × β → E⦄ (hf : Integrable f (μ.prod ν)) :
    Integrable (fun x => ∫ y, f (x, y) ∂ν) μ 
goal: MeasureTheory.Integrable (fun x => ∫ (y : β), f (x, y) ∂ν) μ

full_name: MeasureTheory.Integrable.integral_prod_right
code: 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 
goal: MeasureTheory.Integrable (fun y => ∫ (x : α), f (x, y) ∂μ) ν

full_name: MeasureTheory.integral_prod_swap
code: theorem integral_prod_swap (f : α × β → E) :
    ∫ z, f z.swap ∂ν.prod μ = ∫ z, f z ∂μ.prod ν 
goal: ∫ (z : β × α), f z.swap ∂ν.prod μ = ∫ (z : α × β), f z ∂μ.prod ν

full_name: MeasureTheory.integral_fn_integral_add
code: /-- Integrals commute with addition inside another integral. `F` can be any function. -/
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 
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

full_name: MeasureTheory.integral_fn_integral_sub
code: /-- Integrals commute with subtraction inside another integral.
  `F` can be any measurable function. -/
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 
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

full_name: MeasureTheory.lintegral_fn_integral_sub
code: /-- Integrals commute with subtraction inside a lower Lebesgue integral.
  `F` can be any function. -/
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 
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

full_name: MeasureTheory.integral_integral_add
code: /-- Double integrals commute with addition. -/
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 
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

full_name: MeasureTheory.integral_integral_add'
code: /-- Double integrals commute with addition. This is the version with `(f + g) (x, y)`
  (instead of `f (x, y) + g (x, y)`) in the LHS. -/
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 
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

full_name: MeasureTheory.integral_integral_sub
code: /-- Double integrals commute with subtraction. -/
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 
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

full_name: MeasureTheory.integral_integral_sub'
code: /-- Double integrals commute with subtraction. This is the version with `(f - g) (x, y)`
  (instead of `f (x, y) - g (x, y)`) in the LHS. -/
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 
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

full_name: MeasureTheory.continuous_integral_integral
code: /-- The map that sends an L¹-function `f : α × β → E` to `∫∫f` is continuous. -/
theorem continuous_integral_integral :
    Continuous fun f : α × β →₁[μ.prod ν] E => ∫ x, ∫ y, f (x, y) ∂ν ∂μ 
goal: Continuous fun f => ∫ (x : α), ∫ (y : β), ↑↑f (x, y) ∂ν ∂μ

full_name: MeasureTheory.integral_prod
code: /-- **Fubini's Theorem**: For integrable functions on `α × β`,
  the Bochner integral of `f` is equal to the iterated Bochner integral.
  `integrable_prod_iff` can be useful to show that the function in question in integrable.
  `MeasureTheory.Integrable.integral_prod_right` is useful to show that the inner integral
  of the right-hand side is integrable. -/
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 
goal: ∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ

full_name: MeasureTheory.integral_prod_symm
code: /-- Symmetric version of **Fubini's Theorem**: For integrable functions on `α × β`,
  the Bochner integral of `f` is equal to the iterated Bochner integral.
  This version has the integrals on the right-hand side in the other order. -/
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 
goal: ∫ (z : α × β), f z ∂μ.prod ν = ∫ (y : β), ∫ (x : α), f (x, y) ∂μ ∂ν

full_name: MeasureTheory.integral_integral
code: /-- Reversed version of **Fubini's Theorem**. -/
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 
goal: ∫ (x : α), ∫ (y : β), f x y ∂ν ∂μ = ∫ (z : α × β), f z.1 z.2 ∂μ.prod ν

full_name: MeasureTheory.integral_integral_symm
code: /-- Reversed version of **Fubini's Theorem** (symmetric version). -/
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 
goal: ∫ (x : α), ∫ (y : β), f x y ∂ν ∂μ = ∫ (z : β × α), f z.2 z.1 ∂ν.prod μ

full_name: MeasureTheory.integral_integral_swap
code: /-- Change the order of Bochner integration. -/
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 
goal: ∫ (x : α), ∫ (y : β), f x y ∂ν ∂μ = ∫ (y : β), ∫ (x : α), f x y ∂μ ∂ν

full_name: MeasureTheory.setIntegral_prod
code: /-- **Fubini's Theorem** for set integrals. -/
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 
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

full_name: MeasureTheory.integral_prod_smul
code: 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 
goal: ∫ (z : α × β), f z.1 • g z.2 ∂μ.prod ν = (∫ (x : α), f x ∂μ) • ∫ (y : β), g y ∂ν

full_name: MeasureTheory.integral_prod_mul
code: 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 
goal: ∫ (z : α × β), f z.1 * g z.2 ∂μ.prod ν = (∫ (x : α), f x ∂μ) * ∫ (y : β), g y ∂ν

full_name: MeasureTheory.setIntegral_prod_mul
code: 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 
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

full_name: MeasureTheory.integral_fun_snd
code: theorem integral_fun_snd (f : β → E) : ∫ z, f z.2 ∂μ.prod ν = (μ univ).toReal • ∫ y, f y ∂ν 
goal: ∫ (z : α × β), f z.2 ∂μ.prod ν = (μ Set.univ).toReal • ∫ (y : β), f y ∂ν

full_name: MeasureTheory.integral_fun_fst
code: theorem integral_fun_fst (f : α → E) : ∫ z, f z.1 ∂μ.prod ν = (ν univ).toReal • ∫ x, f x ∂μ 
goal: ∫ (z : α × β), f z.1 ∂μ.prod ν = (ν Set.univ).toReal • ∫ (x : α), f x ∂μ

full_name: MeasureTheory.integral_integral_swap_of_hasCompactSupport
code: /-- A version of *Fubini theorem* for continuous functions with compact support: one may swap
the order of integration with respect to locally finite measures. One does not assume that the
measures are σ-finite, contrary to the usual Fubini theorem. -/
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 
goal: ∫ (x : X), ∫ (y : Y), f x y ∂ν ∂μ = ∫ (y : Y), ∫ (x : X), f x y ∂μ ∂ν