mathlib documentation

measure_theory.prod

The product measure

In this file we define and prove properties about the binary product measure. If α and β have σ-finite measures μ resp. ν then α × β can be equipped with a σ-finite measure μ.prod ν that satisfies (μ.prod ν) s = ∫⁻ x, ν {y | (x, y) ∈ s} ∂μ. We also have (μ.prod ν) (s.prod t) = μ s * ν t, i.e. the measure of a rectangle is the product of the measures of the sides.

We also prove Tonelli's theorem and Fubini's theorem.

Main definition

Main results

Implementation Notes

Many results are proven twice, once for functions in curried form (α → β → γ) and one for functions in uncurried form (α × β → γ). The former often has an assumption measurable (uncurry f), which could be inconvenient to discharge, but for the latter it is more common that the function has to be given explicitly, since Lean cannot synthesize the function by itself. We name the lemmas about the uncurried form with a prime. Tonelli's theorem and Fubini's theorem have a different naming scheme, since the version for the uncurried version is reversed.

Tags

product measure, Fubini's theorem, Tonelli's theorem, Fubini-Tonelli theorem

theorem is_pi_system.prod {α : Type u_1} {β : Type u_3} {C : set (set α)} {D : set (set β)} (hC : is_pi_system C) (hD : is_pi_system D) :

Rectangles formed by π-systems form a π-system.

theorem is_countably_spanning.prod {α : Type u_1} {β : Type u_3} {C : set (set α)} {D : set (set β)} (hC : is_countably_spanning C) (hD : is_countably_spanning D) :

Rectangles of countably spanning sets are countably spanning.

### Measurability

Before we define the product measure, we can talk about the measurability of operations on binary functions. We show that if f is a binary measurable function, then the function that integrates along one of the variables (using either the Lebesgue or Bochner integral) is measurable.

theorem generate_from_prod_eq {α : Type u_1} {β : Type u_2} {C : set (set α)} {D : set (set β)} (hC : is_countably_spanning C) (hD : is_countably_spanning D) :

The product of generated σ-algebras is the one generated by rectangles, if both generating sets are countably spanning.

If C and D generate the σ-algebras on α resp. β, then rectangles formed by C and D generate the σ-algebra on α × β.

The product σ-algebra is generated from boxes, i.e. s.prod t for sets s : set α and t : set β.

theorem is_pi_system_prod {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] :

Rectangles form a π-system.

theorem measurable_measure_prod_mk_left_finite {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {ν : measure_theory.measure β} [measure_theory.finite_measure ν] {s : set × β)} (hs : is_measurable s) :
measurable (λ (x : α), ν (prod.mk x ⁻¹' s))

If ν is a finite measure, and s ⊆ α × β is measurable, then x ↦ ν { y | (x, y) ∈ s } is a measurable function. measurable_measure_prod_mk_left is strictly more general.

theorem measurable_measure_prod_mk_left {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] {s : set × β)} (hs : is_measurable s) :
measurable (λ (x : α), ν (prod.mk x ⁻¹' s))

If ν is a σ-finite measure, and s ⊆ α × β is measurable, then x ↦ ν { y | (x, y) ∈ s } is a measurable function.

theorem measurable_measure_prod_mk_right {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} [measure_theory.sigma_finite μ] {s : set × β)} (hs : is_measurable s) :
measurable (λ (y : β), μ ((λ (x : α), (x, y)) ⁻¹' s))

If μ is a σ-finite measure, and s ⊆ α × β is measurable, then y ↦ μ { x | (x, y) ∈ s } is a measurable function.

theorem measurable.map_prod_mk_right {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} [measure_theory.sigma_finite μ] :
measurable (λ (y : β), (measure_theory.measure.map (λ (x : α), (x, y))) μ)

theorem measurable.lintegral_prod_right' {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] {f : α × βennreal} (hf : measurable f) :
measurable (λ (x : α), ∫⁻ (y : β), f (x, y) ν)

The Lebesgue intergral is measurable. This shows that the integrand of (the right-hand-side of) Tonelli's theorem is measurable.

theorem measurable.lintegral_prod_right {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] {f : α → β → ennreal} (hf : measurable (function.uncurry f)) :
measurable (λ (x : α), ∫⁻ (y : β), f x y ν)

The Lebesgue intergral is measurable. This shows that the integrand of (the right-hand-side of) Tonelli's theorem is measurable. This version has the argument f in curried form.

theorem measurable.lintegral_prod_left' {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} [measure_theory.sigma_finite μ] {f : α × βennreal} (hf : measurable f) :
measurable (λ (y : β), ∫⁻ (x : α), f (x, y) μ)

The Lebesgue intergral is measurable. This shows that the integrand of (the right-hand-side of) the symmetric version of Tonelli's theorem is measurable.

theorem measurable.lintegral_prod_left {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} [measure_theory.sigma_finite μ] {f : α → β → ennreal} (hf : measurable (function.uncurry f)) :
measurable (λ (y : β), ∫⁻ (x : α), f x y μ)

The Lebesgue intergral is measurable. This shows that the integrand of (the right-hand-side of) the symmetric version of Tonelli's theorem is measurable. This version has the argument f in curried form.

theorem is_measurable_integrable {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [opens_measurable_space E] ⦃f : α → β → E (hf : measurable (function.uncurry f)) :

theorem measurable.integral_prod_right {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite ν] ⦃f : α → β → E (hf : measurable (function.uncurry f)) :
measurable (λ (x : α), (y : β), f x y ν)

The Bochner intergral 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 measurable.integral_prod_right' {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite ν] ⦃f : α × β → E⦄ (hf : measurable f) :
measurable (λ (x : α), (y : β), f (x, y) ν)

The Bochner intergral is measurable. This shows that the integrand of (the right-hand-side of) Fubini's theorem is measurable.

theorem measurable.integral_prod_left {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} [normed_group E] [measurable_space E] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] ⦃f : α → β → E (hf : measurable (function.uncurry f)) :
measurable (λ (y : β), (x : α), f x y μ)

The Bochner intergral 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 measurable.integral_prod_left' {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} [normed_group E] [measurable_space E] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] ⦃f : α × β → E⦄ (hf : measurable f) :
measurable (λ (y : β), (x : α), f (x, y) μ)

The Bochner intergral is measurable. This shows that the integrand of (the right-hand-side of) the symmetric version of Fubini's theorem is measurable.

The product measure

def measure_theory.measure.prod {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] (μ : measure_theory.measure α) (ν : measure_theory.measure β) :

The binary product of measures. They are defined for arbitrary measures, but we basically prove all properties under the assumption that at least one of them is σ-finite.

Equations
theorem measure_theory.measure.prod_apply {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] {s : set × β)} (hs : is_measurable s) :
(μ.prod ν) s = ∫⁻ (x : α), ν (prod.mk x ⁻¹' s) μ

@[simp]
theorem measure_theory.measure.prod_prod {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] {s : set α} {t : set β} (hs : is_measurable s) (ht : is_measurable t) :
(μ.prod ν) (s.prod t) = (μ s) * ν t

theorem measure_theory.measure.ae_measure_lt_top {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] {s : set × β)} (hs : is_measurable s) (h2s : (μ.prod ν) s < ) :
∀ᵐ (x : α) ∂μ, ν (prod.mk x ⁻¹' s) <

theorem measure_theory.measure.integrable_measure_prod_mk_left {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] {s : set × β)} (hs : is_measurable s) (h2s : (μ.prod ν) s < ) :
measure_theory.integrable (λ (x : α), (ν (prod.mk x ⁻¹' s)).to_real) μ

theorem measure_theory.measure.measure_prod_null {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] {s : set × β)} (hs : is_measurable s) :
(μ.prod ν) s = 0 (λ (x : α), ν (prod.mk x ⁻¹' s)) =ᵐ[μ] 0

Note: the assumption hs cannot be dropped. For a counterexample, see Walter Rudin Real and Complex Analysis, example (c) in section 8.9.

theorem measure_theory.measure.measure_ae_null_of_prod_null {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] {s : set × β)} (h : (μ.prod ν) s = 0) :
(λ (x : α), ν (prod.mk x ⁻¹' s)) =ᵐ[μ] 0

Note: the converse is not true without assuming that s is measurable. For a counterexample, see Walter Rudin Real and Complex Analysis, example (c) in section 8.9.

theorem measure_theory.measure.ae_ae_of_ae_prod {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] {p : α × β → Prop} (h : ∀ᵐ (z : α × β)μ.prod ν, p z) :
∀ᵐ (x : α) ∂μ, ∀ᵐ (y : β) ∂ν, p (x, y)

Note: the converse is not true. For a counterexample, see Walter Rudin Real and Complex Analysis, example (c) in section 8.9.

def measure_theory.measure.finite_spanning_sets_in.prod {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} {C : set (set α)} {D : set (set β)} (hμ : μ.finite_spanning_sets_in C) (hν : ν.finite_spanning_sets_in D) (hC : ∀ (s : set α), s Cis_measurable s) (hD : ∀ (t : set β), t Dis_measurable t) :

μ.prod ν has finite spanning sets in rectangles of finite spanning sets.

Equations
theorem measure_theory.measure.prod_eq_generate_from {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} {C : set (set α)} {D : set (set β)} (hC : measurable_space.generate_from C = _inst_1) (hD : measurable_space.generate_from D = _inst_3) (h2C : is_pi_system C) (h2D : is_pi_system D) (h3C : μ.finite_spanning_sets_in C) (h3D : ν.finite_spanning_sets_in D) {μν : measure_theory.measure × β)} (h₁ : ∀ (s : set α), s C∀ (t : set β), t Dμν (s.prod t) = (μ s) * ν t) :
μ.prod ν = μν

Measures on a product space are equal the product measure if they are equal on rectangles with as sides sets that generate the corresponding σ-algebras.

theorem measure_theory.measure.prod_eq {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] [measure_theory.sigma_finite μ] {μν : measure_theory.measure × β)} (h : ∀ (s : set α) (t : set β), is_measurable sis_measurable tμν (s.prod t) = (μ s) * ν t) :
μ.prod ν = μν

Measures on a product space are equal to the product measure if they are equal on rectangles.

theorem measure_theory.measure.prod_apply_symm {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] [measure_theory.sigma_finite μ] {s : set × β)} (hs : is_measurable s) :
(μ.prod ν) s = ∫⁻ (y : β), μ ((λ (x : α), (x, y)) ⁻¹' s) ν

The product of specific measures

theorem measure_theory.measure.prod_restrict {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] [measure_theory.sigma_finite μ] {s : set α} {t : set β} (hs : is_measurable s) (ht : is_measurable t) :
(μ.restrict s).prod (ν.restrict t) = (μ.prod ν).restrict (s.prod t)

theorem measure_theory.measure.prod_dirac {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} [measure_theory.sigma_finite μ] (y : β) :

theorem measure_theory.measure.prod_sum {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} [measure_theory.sigma_finite μ] {ι : Type u_2} [fintype ι] (ν : ι → measure_theory.measure β) [∀ (i : ι), measure_theory.sigma_finite (ν i)] :

theorem measure_theory.measure.sum_prod {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] {ι : Type u_2} [fintype ι] (μ : ι → measure_theory.measure α) [∀ (i : ι), measure_theory.sigma_finite (μ i)] :

The Lebesgue integral on a product

theorem measure_theory.lintegral_prod_swap {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] [measure_theory.sigma_finite μ] (f : α × βennreal) (hf : measurable f) :
∫⁻ (z : β × α), f z.swap ν.prod μ = ∫⁻ (z : α × β), f z μ.prod ν

theorem measure_theory.lintegral_prod {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] (f : α × βennreal) (hf : measurable f) :
∫⁻ (z : α × β), f z μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ν μ

Tonelli's Theorem: For ennreal-valued measurable functions on α × β, the integral of f is equal to the iterated integral.

theorem measure_theory.lintegral_prod_symm {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] [measure_theory.sigma_finite μ] (f : α × βennreal) (hf : measurable f) :
∫⁻ (z : α × β), f z μ.prod ν = ∫⁻ (y : β), ∫⁻ (x : α), f (x, y) μ ν

The symmetric verion of Tonelli's Theorem: For ennreal-valued measurable functions on α × β, the integral of f is equal to the iterated integral, in reverse order.

theorem measure_theory.lintegral_lintegral {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] ⦃f : α → β → ennreal (hf : measurable (function.uncurry f)) :
∫⁻ (x : α), ∫⁻ (y : β), f x y ν μ = ∫⁻ (z : α × β), f z.fst z.snd μ.prod ν

The reversed version of Tonelli's Theorem. In this version f is in curried form, which makes it easier for the elaborator to figure out f automatically.

theorem measure_theory.lintegral_lintegral_symm {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] [measure_theory.sigma_finite μ] ⦃f : α → β → ennreal (hf : measurable (function.uncurry f)) :
∫⁻ (x : α), ∫⁻ (y : β), f x y ν μ = ∫⁻ (z : β × α), f z.snd z.fst ν.prod μ

The reversed version of Tonelli's Theorem (symmetric version). In this version f is in curried form, which makes it easier for the elaborator to figure out f automatically.

theorem measure_theory.lintegral_lintegral_swap {α : Type u_1} {β : Type u_3} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [measure_theory.sigma_finite ν] [measure_theory.sigma_finite μ] ⦃f : α → β → ennreal (hf : measurable (function.uncurry f)) :
∫⁻ (x : α), ∫⁻ (y : β), f x y ν μ = ∫⁻ (y : β), ∫⁻ (x : α), f x y μ ν

Change the order of Lebesgue integration.

Integrability on a product

theorem measure_theory.has_finite_integral_prod_iff {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [opens_measurable_space E] ⦃f : α × β → E⦄ (h1f : measurable f) :
measure_theory.has_finite_integral f (μ.prod ν) (∀ᵐ (x : α) ∂μ, measure_theory.has_finite_integral (λ (y : β), f (x, y)) ν) measure_theory.has_finite_integral (λ (x : α), (y : β), f (x, y) ν) μ

theorem measure_theory.integrable_prod_iff {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [opens_measurable_space E] ⦃f : α × β → E⦄ (h1f : measurable f) :
measure_theory.integrable f (μ.prod ν) (∀ᵐ (x : α) ∂μ, measure_theory.integrable (λ (y : β), f (x, y)) ν) measure_theory.integrable (λ (x : α), (y : β), f (x, y) ν) μ

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 measure_theory.integrable_prod_iff' {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [opens_measurable_space E] [measure_theory.sigma_finite μ] ⦃f : α × β → E⦄ (h1f : measurable f) :
measure_theory.integrable f (μ.prod ν) (∀ᵐ (y : β) ∂ν, measure_theory.integrable (λ (x : α), f (x, y)) μ) measure_theory.integrable (λ (y : β), (x : α), f (x, y) μ) ν

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 measure_theory.integrable.prod_left_ae {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [opens_measurable_space E] [measure_theory.sigma_finite μ] ⦃f : α × β → E⦄ (hf : measure_theory.integrable f (μ.prod ν)) :
∀ᵐ (y : β) ∂ν, measure_theory.integrable (λ (x : α), f (x, y)) μ

theorem measure_theory.integrable.prod_right_ae {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [opens_measurable_space E] [measure_theory.sigma_finite μ] ⦃f : α × β → E⦄ (hf : measure_theory.integrable f (μ.prod ν)) :
∀ᵐ (x : α) ∂μ, measure_theory.integrable (λ (y : β), f (x, y)) ν

theorem measure_theory.integrable.integral_norm_prod_left {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [opens_measurable_space E] ⦃f : α × β → E⦄ (hf : measure_theory.integrable f (μ.prod ν)) :
measure_theory.integrable (λ (x : α), (y : β), f (x, y) ν) μ

theorem measure_theory.integrable.integral_norm_prod_right {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [opens_measurable_space E] [measure_theory.sigma_finite μ] ⦃f : α × β → E⦄ (hf : measure_theory.integrable f (μ.prod ν)) :
measure_theory.integrable (λ (y : β), (x : α), f (x, y) μ) ν

The Bochner integral on a product

theorem measure_theory.integral_prod_swap {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] (f : α × β → E) (hf : measurable f) :
(z : β × α), f z.swap ν.prod μ = (z : α × β), f z μ.prod ν

Some rules about the sum/difference of double integrals. They follow from integral_add, but we separate them out as separate lemmas, because they involve quite some steps.

theorem measure_theory.integral_fn_integral_add {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] {E' : Type u_7} [measurable_space E'] [normed_group E'] [borel_space E'] [complete_space E'] [normed_space E'] [topological_space.second_countable_topology E'] ⦃f g : α × β → E⦄ {F : E → E'} (hF : measurable F) (hf : measure_theory.integrable f (μ.prod ν)) (hg : measure_theory.integrable g (μ.prod ν)) :
(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

Integrals commute with addition inside another integral. F can be any measurable function.

theorem measure_theory.integral_fn_integral_sub {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] {E' : Type u_7} [measurable_space E'] [normed_group E'] [borel_space E'] [complete_space E'] [normed_space E'] [topological_space.second_countable_topology E'] ⦃f g : α × β → E⦄ {F : E → E'} (hF : measurable F) (hf : measure_theory.integrable f (μ.prod ν)) (hg : measure_theory.integrable g (μ.prod ν)) :
(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

Integrals commute with subtraction inside another integral. F can be any measurable function.

theorem measure_theory.lintegral_fn_integral_sub {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] ⦃f g : α × β → E⦄ (F : E → ennreal) (hf : measure_theory.integrable f (μ.prod ν)) (hg : measure_theory.integrable g (μ.prod ν)) :
∫⁻ (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

Integrals commute with subtraction inside a lower Lebesgue integral. F can be any function.

theorem measure_theory.integral_integral_add {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] ⦃f g : α × β → E⦄ (hf : measure_theory.integrable f (μ.prod ν)) (hg : measure_theory.integrable g (μ.prod ν)) :
(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

Double integrals commute with addition.

theorem measure_theory.integral_integral_add' {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] ⦃f g : α × β → E⦄ (hf : measure_theory.integrable f (μ.prod ν)) (hg : measure_theory.integrable g (μ.prod ν)) :
(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

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 measure_theory.integral_integral_sub {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] ⦃f g : α × β → E⦄ (hf : measure_theory.integrable f (μ.prod ν)) (hg : measure_theory.integrable g (μ.prod ν)) :
(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

Double integrals commute with subtraction.

theorem measure_theory.integral_integral_sub' {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] ⦃f g : α × β → E⦄ (hf : measure_theory.integrable f (μ.prod ν)) (hg : measure_theory.integrable g (μ.prod ν)) :
(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

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.

The map that sends an L¹-function f : α × β → E to ∫∫f is continuous.

theorem measure_theory.integral_prod {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] (f : α × β → E) (hf : measure_theory.integrable f (μ.prod ν)) :
(z : α × β), f z μ.prod ν = (x : α), (y : β), f (x, y) ν μ

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. measure_theory.integrable.integral_prod_right is useful to show that the inner integral of the right-hand side is integrable.

theorem measure_theory.integral_prod_symm {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] (f : α × β → E) (hf : measure_theory.integrable f (μ.prod ν)) :
(z : α × β), f z μ.prod ν = (y : β), (x : α), f (x, y) μ ν

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 measure_theory.integral_integral {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] {f : α → β → E} (hf : measure_theory.integrable (function.uncurry f) (μ.prod ν)) :
(x : α), (y : β), f x y ν μ = (z : α × β), f z.fst z.snd μ.prod ν

Reversed version of Fubini's Theorem.

theorem measure_theory.integral_integral_symm {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] {f : α → β → E} (hf : measure_theory.integrable (function.uncurry f) (μ.prod ν)) :
(x : α), (y : β), f x y ν μ = (z : β × α), f z.snd z.fst ν.prod μ

Reversed version of Fubini's Theorem (symmetric version).

theorem measure_theory.integral_integral_swap {α : Type u_1} {β : Type u_3} {E : Type u_6} [measurable_space α] [measurable_space β] {μ : measure_theory.measure α} {ν : measure_theory.measure β} [normed_group E] [measurable_space E] [measure_theory.sigma_finite ν] [topological_space.second_countable_topology E] [normed_space E] [complete_space E] [borel_space E] [measure_theory.sigma_finite μ] ⦃f : α → β → E (hf : measure_theory.integrable (function.uncurry f) (μ.prod ν)) :
(x : α), (y : β), f x y ν μ = (y : β), (x : α), f x y μ ν

Change the order of Bochner integration.