The product measure #
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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 ×ˢ 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.
Main definition #
measure_theory.measure.prod: The product of two measures.
Main results #
measure_theory.measure.prod_applystatesμ.prod ν s = ∫⁻ x, ν {y | (x, y) ∈ s} ∂μfor measurables.measure_theory.measure.prod_apply_symmis the reversed version.measure_theory.measure.prod_prodstatesμ.prod ν (s ×ˢ t) = μ s * ν tfor measurable setssandt.measure_theory.lintegral_prod: Tonelli's theorem. It states that for a measurable functionα × β → ℝ≥0∞we have∫⁻ z, f z ∂(μ.prod ν) = ∫⁻ x, ∫⁻ y, f (x, y) ∂ν ∂μ. The version for functionsα → β → ℝ≥0∞is reversed, and calledlintegral_lintegral. Both versions have a variant with_symmappended, where the order of integration is reversed. The lemmameasurable.lintegral_prod_right'states that the inner integral of the right-hand side is measurable.
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 has a different naming scheme, since the version for the uncurried version is
reversed.
Tags #
product measure, 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) :
is_pi_system (set.image2 set.prod C 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.
theorem
generate_from_eq_prod
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space 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_countably_spanning C)
(h2D : is_countably_spanning D) :
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.
theorem
generate_from_prod
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β] :
measurable_space.generate_from (set.image2 set.prod {s : set α | measurable_set s} {t : set β | measurable_set t}) = prod.measurable_space
The product σ-algebra is generated from boxes, i.e. s ×ˢ 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] :
is_pi_system (set.image2 set.prod {s : set α | measurable_set s} {t : set β | measurable_set t})
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.is_finite_measure ν]
{s : set (α × β)}
(hs : measurable_set 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 (α × β)}
(hs : measurable_set 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 (α × β)}
(hs : measurable_set 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_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] :
measurable (λ (x : α), measure_theory.measure.map (prod.mk x) ν)
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_embedding.prod_mk
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{δ : Type u_4}
{mα : measurable_space α}
{mβ : measurable_space β}
{mγ : measurable_space γ}
{mδ : measurable_space δ}
{f : α → β}
{g : γ → δ}
(hg : measurable_embedding g)
(hf : measurable_embedding f) :
measurable_embedding (λ (x : γ × α), (g x.fst, f x.snd))
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 f) :
measurable (λ (x : α), ∫⁻ (y : β), f (x, y) ∂ν)
The Lebesgue integral 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 integral 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 × β → ennreal}
(hf : measurable f) :
measurable (λ (y : β), ∫⁻ (x : α), f (x, y) ∂μ)
The Lebesgue integral 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 integral 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.
The product measure #
@[protected, irreducible]
noncomputable
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) :
measure_theory.measure (α × 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
- μ.prod ν = μ.bind (λ (x : α), measure_theory.measure.map (prod.mk x) ν)
Instances for measure_theory.measure.prod
- measure_theory.measure.prod.is_open_pos_measure
- measure_theory.measure.prod.measure_theory.is_finite_measure
- measure_theory.measure.prod.measure_theory.is_probability_measure
- measure_theory.measure.prod.measure_theory.is_finite_measure_on_compacts
- measure_theory.measure.prod.sigma_finite
- measure_theory.measure.prod.measure.is_mul_left_invariant
- measure_theory.measure.prod.measure.is_add_left_invariant
- measure_theory.measure.prod.measure.is_mul_right_invariant
- measure_theory.measure.prod.measure.is_add_right_invariant
- measure_theory.measure.prod.is_haar_measure
- measure_theory.measure.prod.is_add_haar_measure
@[protected, instance]
noncomputable
def
measure_theory.measure.prod.measure_space
{α : Type u_1}
{β : Type u_2}
[measure_theory.measure_space α]
[measure_theory.measure_space β] :
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 (α × β)}
(hs : measurable_set s) :
@[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) :
The product measure of the product of two sets is the product of their measures. Note that we do not need the sets to be measurable.
@[protected, instance]
def
measure_theory.measure.prod.is_open_pos_measure
{X : Type u_1}
{Y : Type u_2}
[topological_space X]
[topological_space Y]
{m : measurable_space X}
{μ : measure_theory.measure X}
[μ.is_open_pos_measure]
{m' : measurable_space Y}
{ν : measure_theory.measure Y}
[ν.is_open_pos_measure]
[measure_theory.sigma_finite ν] :
(μ.prod ν).is_open_pos_measure
@[protected, instance]
def
measure_theory.measure.prod.measure_theory.is_finite_measure
{α : Type u_1}
{β : Type u_2}
{mα : measurable_space α}
{mβ : measurable_space β}
(μ : measure_theory.measure α)
(ν : measure_theory.measure β)
[measure_theory.is_finite_measure μ]
[measure_theory.is_finite_measure ν] :
@[protected, instance]
def
measure_theory.measure.prod.measure_theory.is_probability_measure
{α : Type u_1}
{β : Type u_2}
{mα : measurable_space α}
{mβ : measurable_space β}
(μ : measure_theory.measure α)
(ν : measure_theory.measure β)
[measure_theory.is_probability_measure μ]
[measure_theory.is_probability_measure ν] :
@[protected, instance]
def
measure_theory.measure.prod.measure_theory.is_finite_measure_on_compacts
{α : Type u_1}
{β : Type u_2}
[topological_space α]
[topological_space β]
{mα : measurable_space α}
{mβ : measurable_space β}
(μ : measure_theory.measure α)
(ν : measure_theory.measure β)
[measure_theory.is_finite_measure_on_compacts μ]
[measure_theory.is_finite_measure_on_compacts ν]
[measure_theory.sigma_finite ν] :
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 (α × β)}
(hs : measurable_set s)
(h2s : ⇑(μ.prod ν) s ≠ ⊤) :
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 (α × β)}
(hs : measurable_set s) :
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 (α × β)}
(h : ⇑(μ.prod ν) s = 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.absolutely_continuous.prod
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
{μ μ' : measure_theory.measure α}
{ν ν' : measure_theory.measure β}
[measure_theory.sigma_finite ν]
[measure_theory.sigma_finite ν']
(h1 : μ.absolutely_continuous μ')
(h2 : ν.absolutely_continuous ν') :
(μ.prod ν).absolutely_continuous (μ'.prod ν')
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) :
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) :
(μ.prod ν).finite_spanning_sets_in (set.image2 set.prod C D)
μ.prod ν has finite spanning sets in rectangles of finite spanning sets.
theorem
measure_theory.measure.quasi_measure_preserving_fst
{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] :
theorem
measure_theory.measure.quasi_measure_preserving_snd
{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] :
@[protected, instance]
def
measure_theory.measure.prod.sigma_finite
{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] :
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 (α × β)}
(h₁ : ∀ (s : set α), s ∈ C → ∀ (t : set β), t ∈ D → ⇑μν (s ×ˢ t) = ⇑μ s * ⇑ν t) :
A measure on a product space equals 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 (α × β)}
(h : ∀ (s : set α) (t : set β), measurable_set s → measurable_set t → ⇑μν (s ×ˢ t) = ⇑μ s * ⇑ν t) :
A measure on a product space equals the product measure if they are equal on rectangles.
theorem
measure_theory.measure.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] :
measure_theory.measure.map prod.swap (μ.prod ν) = ν.prod μ
theorem
measure_theory.measure.measure_preserving_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] :
measure_theory.measure_preserving prod.swap (μ.prod ν) (ν.prod μ)
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 (α × β)}
(hs : measurable_set s) :
theorem
measure_theory.measure.prod_assoc_prod
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{τ : measure_theory.measure γ}
[measure_theory.sigma_finite ν]
[measure_theory.sigma_finite μ]
[measure_theory.sigma_finite τ] :
measure_theory.measure.map ⇑measurable_equiv.prod_assoc ((μ.prod ν).prod τ) = μ.prod (ν.prod τ)
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) :
theorem
measure_theory.measure.restrict_prod_eq_prod_univ
{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) :
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) :
μ.prod (measure_theory.measure.dirac y) = measure_theory.measure.map (λ (x : α), (x, y)) μ
theorem
measure_theory.measure.dirac_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]
(x : α) :
theorem
measure_theory.measure.dirac_prod_dirac
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
{x : α}
{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}
[finite ι]
(ν : ι → measure_theory.measure β)
[∀ (i : ι), measure_theory.sigma_finite (ν i)] :
μ.prod (measure_theory.measure.sum ν) = measure_theory.measure.sum (λ (i : ι), μ.prod (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}
[finite ι]
(μ : ι → measure_theory.measure α)
[∀ (i : ι), measure_theory.sigma_finite (μ i)] :
(measure_theory.measure.sum μ).prod ν = measure_theory.measure.sum (λ (i : ι), (μ i).prod ν)
theorem
measure_theory.measure.prod_add
{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]
(ν' : measure_theory.measure β)
[measure_theory.sigma_finite ν'] :
theorem
measure_theory.measure.add_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]
[measure_theory.sigma_finite μ]
(μ' : measure_theory.measure α)
[measure_theory.sigma_finite μ'] :
@[simp]
theorem
measure_theory.measure.zero_prod
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
(ν : measure_theory.measure β) :
@[simp]
theorem
measure_theory.measure.prod_zero
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
(μ : measure_theory.measure α) :
theorem
measure_theory.measure.map_prod_map
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{δ : Type u_2}
[measurable_space δ]
{f : α → β}
{g : γ → δ}
{μa : measure_theory.measure α}
{μc : measure_theory.measure γ}
(hfa : measure_theory.sigma_finite (measure_theory.measure.map f μa))
(hgc : measure_theory.sigma_finite (measure_theory.measure.map g μc))
(hf : measurable f)
(hg : measurable g) :
(measure_theory.measure.map f μa).prod (measure_theory.measure.map g μc) = measure_theory.measure.map (prod.map f g) (μa.prod μc)
theorem
measure_theory.measure_preserving.skew_product
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{δ : Type u_7}
[measurable_space δ]
{μa : measure_theory.measure α}
{μb : measure_theory.measure β}
{μc : measure_theory.measure γ}
{μd : measure_theory.measure δ}
[measure_theory.sigma_finite μb]
[measure_theory.sigma_finite μd]
{f : α → β}
(hf : measure_theory.measure_preserving f μa μb)
{g : α → γ → δ}
(hgm : measurable (function.uncurry g))
(hg : ∀ᵐ (x : α) ∂μa, measure_theory.measure.map (g x) μc = μd) :
@[protected]
theorem
measure_theory.measure_preserving.prod
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{δ : Type u_7}
[measurable_space δ]
{μa : measure_theory.measure α}
{μb : measure_theory.measure β}
{μc : measure_theory.measure γ}
{μd : measure_theory.measure δ}
[measure_theory.sigma_finite μb]
[measure_theory.sigma_finite μd]
{f : α → β}
{g : γ → δ}
(hf : measure_theory.measure_preserving f μa μb)
(hg : measure_theory.measure_preserving g μc μd) :
measure_theory.measure_preserving (prod.map f g) (μa.prod μc) (μb.prod μd)
If f : α → β sends the measure μa to μb and g : γ → δ sends the measure μc to μd,
then prod.map f g sends μa.prod μc to μb.prod μd.
theorem
measure_theory.quasi_measure_preserving.prod_of_right
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{f : α × β → γ}
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{τ : measure_theory.measure γ}
(hf : measurable f)
[measure_theory.sigma_finite ν]
(h2f : ∀ᵐ (x : α) ∂μ, measure_theory.measure.quasi_measure_preserving (λ (y : β), f (x, y)) ν τ) :
theorem
measure_theory.quasi_measure_preserving.prod_of_left
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{f : α × β → γ}
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
{τ : measure_theory.measure γ}
(hf : measurable f)
[measure_theory.sigma_finite μ]
[measure_theory.sigma_finite ν]
(h2f : ∀ᵐ (y : β) ∂ν, measure_theory.measure.quasi_measure_preserving (λ (x : α), f (x, y)) μ τ) :
theorem
ae_measurable.prod_swap
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
[measure_theory.sigma_finite μ]
[measure_theory.sigma_finite ν]
{f : β × α → γ}
(hf : ae_measurable f (ν.prod μ)) :
ae_measurable (λ (z : α × β), f z.swap) (μ.prod ν)
theorem
ae_measurable.fst
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
[measure_theory.sigma_finite ν]
{f : α → γ}
(hf : ae_measurable f μ) :
ae_measurable (λ (z : α × β), f z.fst) (μ.prod ν)
theorem
ae_measurable.snd
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{μ : measure_theory.measure α}
{ν : measure_theory.measure β}
[measure_theory.sigma_finite ν]
{f : β → γ}
(hf : ae_measurable f ν) :
ae_measurable (λ (z : α × β), f z.snd) (μ.prod ν)
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 × β → ennreal)
(hf : ae_measurable f (μ.prod ν)) :
theorem
measure_theory.lintegral_prod_of_measurable
{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 f) :
Tonelli's Theorem: For ℝ≥0∞-valued measurable functions on α × β,
the integral of f is equal to the iterated integral.
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 × β → ennreal)
(hf : ae_measurable f (μ.prod ν)) :
Tonelli's Theorem: For ℝ≥0∞-valued almost everywhere 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 × β → ennreal)
(hf : ae_measurable f (μ.prod ν)) :
The symmetric verion of Tonelli's Theorem: For ℝ≥0∞-valued almost everywhere measurable
functions on α × β, the integral of f is equal to the iterated integral, in reverse order.
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 × β → ennreal)
(hf : measurable f) :
The symmetric verion of Tonelli's Theorem: For ℝ≥0∞-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 : ae_measurable (function.uncurry f) (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 : ae_measurable (function.uncurry f) (μ.prod ν)) :
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 : ae_measurable (function.uncurry f) (μ.prod ν)) :
Change the order of Lebesgue integration.
theorem
measure_theory.lintegral_prod_mul
{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 → ennreal}
{g : β → ennreal}
(hf : ae_measurable f μ)
(hg : ae_measurable g ν) :
Marginals of a measure defined on a product #
noncomputable
def
measure_theory.measure.fst
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
(ρ : measure_theory.measure (α × β)) :
Marginal measure on α obtained from a measure ρ on α × β, defined by ρ.map prod.fst.
Equations
Instances for measure_theory.measure.fst
theorem
measure_theory.measure.fst_apply
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
{ρ : measure_theory.measure (α × β)}
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.measure.fst_univ
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
{ρ : measure_theory.measure (α × β)} :
@[protected, instance]
def
measure_theory.measure.fst.measure_theory.is_finite_measure
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
{ρ : measure_theory.measure (α × β)}
[measure_theory.is_finite_measure ρ] :
@[protected, instance]
def
measure_theory.measure.fst.measure_theory.is_probability_measure
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
{ρ : measure_theory.measure (α × β)}
[measure_theory.is_probability_measure ρ] :
theorem
measure_theory.measure.fst_map_prod_mk₀
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{X : α → β}
{Y : α → γ}
{μ : measure_theory.measure α}
(hX : ae_measurable X μ)
(hY : ae_measurable Y μ) :
(measure_theory.measure.map (λ (a : α), (X a, Y a)) μ).fst = measure_theory.measure.map X μ
theorem
measure_theory.measure.fst_map_prod_mk
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{X : α → β}
{Y : α → γ}
{μ : measure_theory.measure α}
(hX : measurable X)
(hY : measurable Y) :
(measure_theory.measure.map (λ (a : α), (X a, Y a)) μ).fst = measure_theory.measure.map X μ
noncomputable
def
measure_theory.measure.snd
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
(ρ : measure_theory.measure (α × β)) :
Marginal measure on β obtained from a measure on ρ α × β, defined by ρ.map prod.snd.
Equations
Instances for measure_theory.measure.snd
theorem
measure_theory.measure.snd_apply
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
{ρ : measure_theory.measure (α × β)}
{s : set β}
(hs : measurable_set s) :
theorem
measure_theory.measure.snd_univ
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
{ρ : measure_theory.measure (α × β)} :
@[protected, instance]
def
measure_theory.measure.snd.measure_theory.is_finite_measure
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
{ρ : measure_theory.measure (α × β)}
[measure_theory.is_finite_measure ρ] :
@[protected, instance]
def
measure_theory.measure.snd.measure_theory.is_probability_measure
{α : Type u_1}
{β : Type u_3}
[measurable_space α]
[measurable_space β]
{ρ : measure_theory.measure (α × β)}
[measure_theory.is_probability_measure ρ] :
theorem
measure_theory.measure.snd_map_prod_mk₀
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{X : α → β}
{Y : α → γ}
{μ : measure_theory.measure α}
(hX : ae_measurable X μ)
(hY : ae_measurable Y μ) :
(measure_theory.measure.map (λ (a : α), (X a, Y a)) μ).snd = measure_theory.measure.map Y μ
theorem
measure_theory.measure.snd_map_prod_mk
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[measurable_space α]
[measurable_space β]
[measurable_space γ]
{X : α → β}
{Y : α → γ}
{μ : measure_theory.measure α}
(hX : measurable X)
(hY : measurable Y) :
(measure_theory.measure.map (λ (a : α), (X a, Y a)) μ).snd = measure_theory.measure.map Y μ