The product measure #
In this file we define and prove properties about the binary product measure. If α and β have
s-finite measures μ resp. ν then α × β can be equipped with a s-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 #
MeasureTheory.Measure.prod: The product of two measures.
Main results #
MeasureTheory.Measure.prod_applystatesμ.prod ν s = ∫⁻ x, ν {y | (x, y) ∈ s} ∂μfor measurables.MeasureTheory.Measure.prod_apply_symmis the reversed version.MeasureTheory.Measure.prod_prodstatesμ.prod ν (s ×ˢ t) = μ s * ν tfor measurable setssandt.MeasureTheory.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
IsPiSystem.prod
{α : Type u_1}
{β : Type u_3}
{C : Set (Set α)}
{D : Set (Set β)}
(hC : IsPiSystem C)
(hD : IsPiSystem D)
:
IsPiSystem (Set.image2 (fun (x : Set α) (x_1 : Set β) => x ×ˢ x_1) C D)
Rectangles formed by π-systems form a π-system.
theorem
IsCountablySpanning.prod
{α : Type u_1}
{β : Type u_3}
{C : Set (Set α)}
{D : Set (Set β)}
(hC : IsCountablySpanning C)
(hD : IsCountablySpanning D)
:
IsCountablySpanning (Set.image2 (fun (x : Set α) (x_1 : Set β) => x ×ˢ x_1) C 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
generateFrom_prod_eq
{α : Type u_7}
{β : Type u_8}
{C : Set (Set α)}
{D : Set (Set β)}
(hC : IsCountablySpanning C)
(hD : IsCountablySpanning D)
:
Prod.instMeasurableSpace = MeasurableSpace.generateFrom (Set.image2 (fun (x : Set α) (x_1 : Set β) => x ×ˢ x_1) C D)
The product of generated σ-algebras is the one generated by rectangles, if both generating sets are countably spanning.
theorem
generateFrom_eq_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{C : Set (Set α)}
{D : Set (Set β)}
(hC : MeasurableSpace.generateFrom C = inst✝¹)
(hD : MeasurableSpace.generateFrom D = inst✝)
(h2C : IsCountablySpanning C)
(h2D : IsCountablySpanning D)
:
MeasurableSpace.generateFrom (Set.image2 (fun (x : Set α) (x_1 : Set β) => x ×ˢ x_1) C D) = Prod.instMeasurableSpace
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
generateFrom_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
:
MeasurableSpace.generateFrom
(Set.image2 (fun (x : Set α) (x_1 : Set β) => x ×ˢ x_1) {s : Set α | MeasurableSet s}
{t : Set β | MeasurableSet t}) = Prod.instMeasurableSpace
The product σ-algebra is generated from boxes, i.e. s ×ˢ t for sets s : Set α and
t : Set β.
theorem
isPiSystem_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
:
IsPiSystem
(Set.image2 (fun (x : Set α) (x_1 : Set β) => x ×ˢ x_1) {s : Set α | MeasurableSet s} {t : Set β | MeasurableSet t})
Rectangles form a π-system.
theorem
measurable_measure_prod_mk_left_finite
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ν : MeasureTheory.Measure β}
[MeasureTheory.IsFiniteMeasure ν]
{s : Set (α × β)}
(hs : MeasurableSet s)
:
Measurable fun (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}
[MeasurableSpace α]
[MeasurableSpace β]
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{s : Set (α × β)}
(hs : MeasurableSet s)
:
Measurable fun (x : α) => ↑↑ν (Prod.mk x ⁻¹' s)
If ν is an s-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}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
[MeasureTheory.SFinite μ]
{s : Set (α × β)}
(hs : MeasurableSet s)
:
Measurable fun (y : β) => ↑↑μ ((fun (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}
[MeasurableSpace α]
[MeasurableSpace β]
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
:
Measurable fun (x : α) => MeasureTheory.Measure.map (Prod.mk x) ν
theorem
Measurable.map_prod_mk_right
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
[MeasureTheory.SFinite μ]
:
Measurable fun (y : β) => MeasureTheory.Measure.map (fun (x : α) => (x, y)) μ
theorem
MeasurableEmbedding.prod_mk
{α : Type u_7}
{β : Type u_8}
{γ : Type u_9}
{δ : Type u_10}
{mα : MeasurableSpace α}
{mβ : MeasurableSpace β}
{mγ : MeasurableSpace γ}
{mδ : MeasurableSpace δ}
{f : α → β}
{g : γ → δ}
(hg : MeasurableEmbedding g)
(hf : MeasurableEmbedding f)
:
MeasurableEmbedding fun (x : γ × α) => (g x.1, f x.2)
theorem
MeasurableEmbedding.prod_mk_left
{α : Type u_1}
[MeasurableSpace α]
{β : Type u_7}
{γ : Type u_8}
[MeasurableSingletonClass α]
{mβ : MeasurableSpace β}
{mγ : MeasurableSpace γ}
(x : α)
{f : γ → β}
(hf : MeasurableEmbedding f)
:
MeasurableEmbedding fun (y : γ) => (x, f y)
theorem
measurableEmbedding_prod_mk_left
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSingletonClass α]
(x : α)
:
theorem
MeasurableEmbedding.prod_mk_right
{α : Type u_1}
[MeasurableSpace α]
{β : Type u_7}
{γ : Type u_8}
[MeasurableSingletonClass α]
{mβ : MeasurableSpace β}
{mγ : MeasurableSpace γ}
{f : γ → β}
(hf : MeasurableEmbedding f)
(x : α)
:
MeasurableEmbedding fun (y : γ) => (f y, x)
theorem
measurableEmbedding_prod_mk_right
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSingletonClass α]
(x : α)
:
MeasurableEmbedding fun (y : β) => (y, x)
theorem
Measurable.lintegral_prod_right'
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{f : α × β → ENNReal}
:
Measurable f → Measurable fun (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}
[MeasurableSpace α]
[MeasurableSpace β]
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{f : α → β → ENNReal}
(hf : Measurable (Function.uncurry f))
:
Measurable fun (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}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
[MeasureTheory.SFinite μ]
{f : α × β → ENNReal}
(hf : Measurable f)
:
Measurable fun (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}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
[MeasureTheory.SFinite μ]
{f : α → β → ENNReal}
(hf : Measurable (Function.uncurry f))
:
Measurable fun (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 #
@[irreducible]
def
MeasureTheory.Measure.prod
{α : Type u_7}
{β : Type u_8}
[MeasurableSpace α]
[MeasurableSpace β]
(μ : MeasureTheory.Measure α)
(ν : MeasureTheory.Measure β)
:
MeasureTheory.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 s-finite.
Instances For
theorem
MeasureTheory.Measure.prod_def
{α : Type u_7}
{β : Type u_8}
[MeasurableSpace α]
[MeasurableSpace β]
(μ : MeasureTheory.Measure α)
(ν : MeasureTheory.Measure β)
:
MeasureTheory.Measure.prod μ ν = MeasureTheory.Measure.bind μ fun (x : α) => MeasureTheory.Measure.map (Prod.mk x) ν
instance
MeasureTheory.Measure.prod.measureSpace
{α : Type u_7}
{β : Type u_8}
[MeasureTheory.MeasureSpace α]
[MeasureTheory.MeasureSpace β]
:
MeasureTheory.MeasureSpace (α × β)
Equations
- MeasureTheory.Measure.prod.measureSpace = MeasureTheory.MeasureSpace.mk (MeasureTheory.Measure.prod MeasureTheory.volume MeasureTheory.volume)
theorem
MeasureTheory.Measure.volume_eq_prod
(α : Type u_7)
(β : Type u_8)
[MeasureTheory.MeasureSpace α]
[MeasureTheory.MeasureSpace β]
:
MeasureTheory.volume = MeasureTheory.Measure.prod MeasureTheory.volume MeasureTheory.volume
theorem
MeasureTheory.Measure.prod_apply
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{s : Set (α × β)}
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.prod μ ν) s = ∫⁻ (x : α), ↑↑ν (Prod.mk x ⁻¹' s) ∂μ
@[simp]
theorem
MeasureTheory.Measure.prod_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
(s : Set α)
(t : Set β)
:
↑↑(MeasureTheory.Measure.prod μ ν) (s ×ˢ t) = ↑↑μ s * ↑↑ν t
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.
@[simp]
theorem
MeasureTheory.Measure.map_fst_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
:
MeasureTheory.Measure.map Prod.fst (MeasureTheory.Measure.prod μ ν) = ↑↑ν Set.univ • μ
@[simp]
theorem
MeasureTheory.Measure.map_snd_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
:
MeasureTheory.Measure.map Prod.snd (MeasureTheory.Measure.prod μ ν) = ↑↑μ Set.univ • ν
instance
MeasureTheory.Measure.prod.instIsOpenPosMeasure
{X : Type u_7}
{Y : Type u_8}
[TopologicalSpace X]
[TopologicalSpace Y]
{m : MeasurableSpace X}
{μ : MeasureTheory.Measure X}
[MeasureTheory.Measure.IsOpenPosMeasure μ]
{m' : MeasurableSpace Y}
{ν : MeasureTheory.Measure Y}
[MeasureTheory.Measure.IsOpenPosMeasure ν]
[MeasureTheory.SFinite ν]
:
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.instIsOpenPosMeasureProdInstTopologicalSpaceProdToMeasurableSpaceMeasureSpaceVolume
{X : Type u_7}
{Y : Type u_8}
[TopologicalSpace X]
[MeasureTheory.MeasureSpace X]
[MeasureTheory.Measure.IsOpenPosMeasure MeasureTheory.volume]
[TopologicalSpace Y]
[MeasureTheory.MeasureSpace Y]
[MeasureTheory.Measure.IsOpenPosMeasure MeasureTheory.volume]
[MeasureTheory.SFinite MeasureTheory.volume]
:
MeasureTheory.Measure.IsOpenPosMeasure MeasureTheory.volume
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.prod.instIsFiniteMeasure
{α : Type u_7}
{β : Type u_8}
{mα : MeasurableSpace α}
{mβ : MeasurableSpace β}
(μ : MeasureTheory.Measure α)
(ν : MeasureTheory.Measure β)
[MeasureTheory.IsFiniteMeasure μ]
[MeasureTheory.IsFiniteMeasure ν]
:
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.instIsFiniteMeasureProdToMeasurableSpaceMeasureSpaceVolume
{α : Type u_7}
{β : Type u_8}
[MeasureTheory.MeasureSpace α]
[MeasureTheory.MeasureSpace β]
[MeasureTheory.IsFiniteMeasure MeasureTheory.volume]
[MeasureTheory.IsFiniteMeasure MeasureTheory.volume]
:
MeasureTheory.IsFiniteMeasure MeasureTheory.volume
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.prod.instIsProbabilityMeasure
{α : Type u_7}
{β : Type u_8}
{mα : MeasurableSpace α}
{mβ : MeasurableSpace β}
(μ : MeasureTheory.Measure α)
(ν : MeasureTheory.Measure β)
[MeasureTheory.IsProbabilityMeasure μ]
[MeasureTheory.IsProbabilityMeasure ν]
:
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.instIsProbabilityMeasureProdToMeasurableSpaceMeasureSpaceVolume
{α : Type u_7}
{β : Type u_8}
[MeasureTheory.MeasureSpace α]
[MeasureTheory.MeasureSpace β]
[MeasureTheory.IsProbabilityMeasure MeasureTheory.volume]
[MeasureTheory.IsProbabilityMeasure MeasureTheory.volume]
:
MeasureTheory.IsProbabilityMeasure MeasureTheory.volume
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.prod.instIsFiniteMeasureOnCompacts
{α : Type u_7}
{β : Type u_8}
[TopologicalSpace α]
[TopologicalSpace β]
{mα : MeasurableSpace α}
{mβ : MeasurableSpace β}
(μ : MeasureTheory.Measure α)
(ν : MeasureTheory.Measure β)
[MeasureTheory.IsFiniteMeasureOnCompacts μ]
[MeasureTheory.IsFiniteMeasureOnCompacts ν]
[MeasureTheory.SFinite ν]
:
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.instIsFiniteMeasureOnCompactsProdToMeasurableSpaceMeasureSpaceInstTopologicalSpaceProdVolume
{X : Type u_7}
{Y : Type u_8}
[TopologicalSpace X]
[MeasureTheory.MeasureSpace X]
[MeasureTheory.IsFiniteMeasureOnCompacts MeasureTheory.volume]
[TopologicalSpace Y]
[MeasureTheory.MeasureSpace Y]
[MeasureTheory.IsFiniteMeasureOnCompacts MeasureTheory.volume]
[MeasureTheory.SFinite MeasureTheory.volume]
:
MeasureTheory.IsFiniteMeasureOnCompacts MeasureTheory.volume
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.prod.instNoAtoms_fst
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
[MeasureTheory.NoAtoms μ]
:
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.prod.instNoAtoms_snd
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
[MeasureTheory.NoAtoms ν]
:
Equations
- ⋯ = ⋯
theorem
MeasureTheory.Measure.ae_measure_lt_top
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{s : Set (α × β)}
(hs : MeasurableSet s)
(h2s : ↑↑(MeasureTheory.Measure.prod μ ν) s ≠ ⊤)
:
theorem
MeasureTheory.Measure.measure_prod_null
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{s : Set (α × β)}
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.prod μ ν) s = 0 ↔ (fun (x : α) => ↑↑ν (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae 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
MeasureTheory.Measure.measure_ae_null_of_prod_null
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{s : Set (α × β)}
(h : ↑↑(MeasureTheory.Measure.prod μ ν) s = 0)
:
(fun (x : α) => ↑↑ν (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae 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
MeasureTheory.Measure.AbsolutelyContinuous.prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{μ' : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
{ν' : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
[MeasureTheory.SFinite ν']
(h1 : MeasureTheory.Measure.AbsolutelyContinuous μ μ')
(h2 : MeasureTheory.Measure.AbsolutelyContinuous ν ν')
:
theorem
MeasureTheory.Measure.ae_ae_of_ae_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{p : α × β → Prop}
(h : ∀ᵐ (z : α × β) ∂MeasureTheory.Measure.prod μ 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. It is true if the set is
measurable, see ae_prod_mem_iff_ae_ae_mem.
theorem
MeasureTheory.Measure.ae_ae_eq_curry_of_prod
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{f : α × β → γ}
{g : α × β → γ}
(h : f =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.prod μ ν)] g)
:
∀ᵐ (x : α) ∂μ, Function.curry f x =ᶠ[MeasureTheory.Measure.ae ν] Function.curry g x
theorem
MeasureTheory.Measure.ae_ae_eq_of_ae_eq_uncurry
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{f : α → β → γ}
{g : α → β → γ}
(h : Function.uncurry f =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.prod μ ν)] Function.uncurry g)
:
∀ᵐ (x : α) ∂μ, f x =ᶠ[MeasureTheory.Measure.ae ν] g x
theorem
MeasureTheory.Measure.ae_prod_mem_iff_ae_ae_mem
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{s : Set (α × β)}
(hs : MeasurableSet s)
:
(∀ᵐ (z : α × β) ∂MeasureTheory.Measure.prod μ ν, z ∈ s) ↔ ∀ᵐ (x : α) ∂μ, ∀ᵐ (y : β) ∂ν, (x, y) ∈ s
theorem
MeasureTheory.Measure.quasiMeasurePreserving_fst
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
:
MeasureTheory.Measure.QuasiMeasurePreserving Prod.fst (MeasureTheory.Measure.prod μ ν) μ
theorem
MeasureTheory.Measure.quasiMeasurePreserving_snd
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
:
MeasureTheory.Measure.QuasiMeasurePreserving Prod.snd (MeasureTheory.Measure.prod μ ν) ν
theorem
MeasureTheory.Measure.set_prod_ae_eq
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{s : Set α}
{s' : Set α}
{t : Set β}
{t' : Set β}
(hs : s =ᶠ[MeasureTheory.Measure.ae μ] s')
(ht : t =ᶠ[MeasureTheory.Measure.ae ν] t')
:
s ×ˢ t =ᶠ[MeasureTheory.Measure.ae (MeasureTheory.Measure.prod μ ν)] s' ×ˢ t'
theorem
MeasureTheory.Measure.measure_prod_compl_eq_zero
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{s : Set α}
{t : Set β}
(s_ae_univ : ↑↑μ sᶜ = 0)
(t_ae_univ : ↑↑ν tᶜ = 0)
:
↑↑(MeasureTheory.Measure.prod μ ν) (s ×ˢ t)ᶜ = 0
theorem
MeasureTheory.NullMeasurableSet.prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{s : Set α}
{t : Set β}
(s_mble : MeasureTheory.NullMeasurableSet s μ)
(t_mble : MeasureTheory.NullMeasurableSet t ν)
:
theorem
MeasureTheory.NullMeasurableSet.right_of_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{s : Set α}
{t : Set β}
(h : MeasureTheory.NullMeasurableSet (s ×ˢ t) (MeasureTheory.Measure.prod μ ν))
(hs : ↑↑μ s ≠ 0)
:
If s ×ˢ t is a null measurable set and μ s ≠ 0, then t is a null measurable set.
theorem
MeasureTheory.NullMeasurableSet.of_preimage_snd
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
[NeZero μ]
{t : Set β}
(h : MeasureTheory.NullMeasurableSet (Prod.snd ⁻¹' t) (MeasureTheory.Measure.prod μ ν))
:
If Prod.snd ⁻¹' t is a null measurable set and μ ≠ 0, then t is a null measurable set.
theorem
MeasureTheory.Measure.nullMeasurableSet_preimage_snd
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
[NeZero μ]
{t : Set β}
:
MeasureTheory.NullMeasurableSet (Prod.snd ⁻¹' t) (MeasureTheory.Measure.prod μ ν) ↔ MeasureTheory.NullMeasurableSet t ν
Prod.snd ⁻¹' t is null measurable w.r.t. μ.prod ν iff t is null measurable w.r.t. ν
provided that μ ≠ 0.
theorem
MeasureTheory.Measure.nullMeasurable_comp_snd
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
[NeZero μ]
{f : β → γ}
:
MeasureTheory.NullMeasurable (f ∘ Prod.snd) (MeasureTheory.Measure.prod μ ν) ↔ MeasureTheory.NullMeasurable f ν
noncomputable def
MeasureTheory.Measure.FiniteSpanningSetsIn.prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
{C : Set (Set α)}
{D : Set (Set β)}
(hμ : MeasureTheory.Measure.FiniteSpanningSetsIn μ C)
(hν : MeasureTheory.Measure.FiniteSpanningSetsIn ν D)
:
MeasureTheory.Measure.FiniteSpanningSetsIn (MeasureTheory.Measure.prod μ ν)
(Set.image2 (fun (x : Set α) (x_1 : Set β) => x ×ˢ x_1) C D)
μ.prod ν has finite spanning sets in rectangles of finite spanning sets.
Equations
- MeasureTheory.Measure.FiniteSpanningSetsIn.prod hμ hν = { set := fun (n : ℕ) => hμ.set (Nat.unpair n).1 ×ˢ hν.set (Nat.unpair n).2, set_mem := ⋯, finite := ⋯, spanning := ⋯ }
Instances For
theorem
MeasureTheory.Measure.prod_sum_left
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ι : Type u_7}
(m : ι → MeasureTheory.Measure α)
(μ : MeasureTheory.Measure β)
[MeasureTheory.SFinite μ]
:
MeasureTheory.Measure.prod (MeasureTheory.Measure.sum m) μ = MeasureTheory.Measure.sum fun (i : ι) => MeasureTheory.Measure.prod (m i) μ
theorem
MeasureTheory.Measure.prod_sum_right
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ι' : Type u_7}
[Countable ι']
(m : MeasureTheory.Measure α)
(m' : ι' → MeasureTheory.Measure β)
[∀ (n : ι'), MeasureTheory.SFinite (m' n)]
:
MeasureTheory.Measure.prod m (MeasureTheory.Measure.sum m') = MeasureTheory.Measure.sum fun (p : ι') => MeasureTheory.Measure.prod m (m' p)
theorem
MeasureTheory.Measure.prod_sum
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ι : Type u_7}
{ι' : Type u_8}
[Countable ι']
(m : ι → MeasureTheory.Measure α)
(m' : ι' → MeasureTheory.Measure β)
[∀ (n : ι'), MeasureTheory.SFinite (m' n)]
:
MeasureTheory.Measure.prod (MeasureTheory.Measure.sum m) (MeasureTheory.Measure.sum m') = MeasureTheory.Measure.sum fun (p : ι × ι') => MeasureTheory.Measure.prod (m p.1) (m' p.2)
instance
MeasureTheory.Measure.prod.instSigmaFinite
{α : Type u_7}
{β : Type u_8}
:
∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SigmaFinite μ] {x_1 : MeasurableSpace β}
{ν : MeasureTheory.Measure β} [inst : MeasureTheory.SigmaFinite ν],
MeasureTheory.SigmaFinite (MeasureTheory.Measure.prod μ ν)
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.prod.instSFinite
{α : Type u_7}
{β : Type u_8}
:
∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.SFinite μ] {x_1 : MeasurableSpace β}
{ν : MeasureTheory.Measure β} [inst : MeasureTheory.SFinite ν], MeasureTheory.SFinite (MeasureTheory.Measure.prod μ ν)
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.instSigmaFiniteProdToMeasurableSpaceMeasureSpaceVolume
{α : Type u_7}
{β : Type u_8}
[MeasureTheory.MeasureSpace α]
[MeasureTheory.SigmaFinite MeasureTheory.volume]
[MeasureTheory.MeasureSpace β]
[MeasureTheory.SigmaFinite MeasureTheory.volume]
:
MeasureTheory.SigmaFinite MeasureTheory.volume
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.instSFiniteProdToMeasurableSpaceMeasureSpaceVolume
{α : Type u_7}
{β : Type u_8}
[MeasureTheory.MeasureSpace α]
[MeasureTheory.SFinite MeasureTheory.volume]
[MeasureTheory.MeasureSpace β]
[MeasureTheory.SFinite MeasureTheory.volume]
:
MeasureTheory.SFinite MeasureTheory.volume
Equations
- ⋯ = ⋯
theorem
MeasureTheory.Measure.prod_eq_generateFrom
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
{C : Set (Set α)}
{D : Set (Set β)}
(hC : MeasurableSpace.generateFrom C = inst✝¹)
(hD : MeasurableSpace.generateFrom D = inst✝)
(h2C : IsPiSystem C)
(h2D : IsPiSystem D)
(h3C : MeasureTheory.Measure.FiniteSpanningSetsIn μ C)
(h3D : MeasureTheory.Measure.FiniteSpanningSetsIn ν D)
{μν : MeasureTheory.Measure (α × β)}
(h₁ : ∀ s ∈ C, ∀ t ∈ D, ↑↑μν (s ×ˢ t) = ↑↑μ s * ↑↑ν t)
:
MeasureTheory.Measure.prod μ ν = μν
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
MeasureTheory.Measure.prod_eq
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
[MeasureTheory.SigmaFinite μ]
{ν : MeasureTheory.Measure β}
[MeasureTheory.SigmaFinite ν]
{μν : MeasureTheory.Measure (α × β)}
(h : ∀ (s : Set α) (t : Set β), MeasurableSet s → MeasurableSet t → ↑↑μν (s ×ˢ t) = ↑↑μ s * ↑↑ν t)
:
MeasureTheory.Measure.prod μ ν = μν
A measure on a product space equals the product measure of sigma-finite measures if they are equal on rectangles.
theorem
MeasureTheory.Measure.prod_swap
{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]
:
MeasureTheory.Measure.map Prod.swap (MeasureTheory.Measure.prod μ ν) = MeasureTheory.Measure.prod ν μ
theorem
MeasureTheory.Measure.measurePreserving_swap
{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]
:
MeasureTheory.MeasurePreserving Prod.swap (MeasureTheory.Measure.prod μ ν) (MeasureTheory.Measure.prod ν μ)
theorem
MeasureTheory.Measure.prod_apply_symm
{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]
{s : Set (α × β)}
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.prod μ ν) s = ∫⁻ (y : β), ↑↑μ ((fun (x : α) => (x, y)) ⁻¹' s) ∂ν
theorem
MeasureTheory.NullMeasurableSet.left_of_prod
{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]
{s : Set α}
{t : Set β}
(h : MeasureTheory.NullMeasurableSet (s ×ˢ t) (MeasureTheory.Measure.prod μ ν))
(ht : ↑↑ν t ≠ 0)
:
If s ×ˢ t is a null measurable set and ν t ≠ 0, then s is a null measurable set.
theorem
MeasureTheory.NullMeasurableSet.of_preimage_fst
{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]
[NeZero ν]
{s : Set α}
(h : MeasureTheory.NullMeasurableSet (Prod.fst ⁻¹' s) (MeasureTheory.Measure.prod μ ν))
:
If Prod.fst ⁻¹' s is a null measurable set and ν ≠ 0, then s is a null measurable set.
theorem
MeasureTheory.Measure.nullMeasurableSet_preimage_fst
{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]
[NeZero ν]
{s : Set α}
:
MeasureTheory.NullMeasurableSet (Prod.fst ⁻¹' s) (MeasureTheory.Measure.prod μ ν) ↔ MeasureTheory.NullMeasurableSet s μ
Prod.fst ⁻¹' s is null measurable w.r.t. μ.prod ν iff s is null measurable w.r.t. μ
provided that ν ≠ 0.
theorem
MeasureTheory.Measure.nullMeasurable_comp_fst
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
[MeasureTheory.SFinite μ]
[NeZero ν]
{f : α → γ}
:
MeasureTheory.NullMeasurable (f ∘ Prod.fst) (MeasureTheory.Measure.prod μ ν) ↔ MeasureTheory.NullMeasurable f μ
theorem
MeasureTheory.Measure.nullMeasurableSet_prod_of_ne_zero
{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]
{s : Set α}
{t : Set β}
(hs : ↑↑μ s ≠ 0)
(ht : ↑↑ν t ≠ 0)
:
The product of two non-null sets is null measurable if and only if both of them are null measurable.
theorem
MeasureTheory.Measure.nullMeasurableSet_prod
{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]
{s : Set α}
{t : Set β}
:
MeasureTheory.NullMeasurableSet (s ×ˢ t) (MeasureTheory.Measure.prod μ ν) ↔ MeasureTheory.NullMeasurableSet s μ ∧ MeasureTheory.NullMeasurableSet t ν ∨ ↑↑μ s = 0 ∨ ↑↑ν t = 0
The product of two sets is null measurable if and only if both of them are null measurable or one of them has measure zero.
theorem
MeasureTheory.Measure.prodAssoc_prod
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
{τ : MeasureTheory.Measure γ}
[MeasureTheory.SFinite ν]
[MeasureTheory.SFinite μ]
[MeasureTheory.SFinite τ]
:
MeasureTheory.Measure.map (⇑MeasurableEquiv.prodAssoc) (MeasureTheory.Measure.prod (MeasureTheory.Measure.prod μ ν) τ) = MeasureTheory.Measure.prod μ (MeasureTheory.Measure.prod ν τ)
The product of specific measures #
theorem
MeasureTheory.Measure.prod_restrict
{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]
(s : Set α)
(t : Set β)
:
MeasureTheory.Measure.prod (μ.restrict s) (ν.restrict t) = (MeasureTheory.Measure.prod μ ν).restrict (s ×ˢ t)
theorem
MeasureTheory.Measure.restrict_prod_eq_prod_univ
{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]
(s : Set α)
:
MeasureTheory.Measure.prod (μ.restrict s) ν = (MeasureTheory.Measure.prod μ ν).restrict (s ×ˢ Set.univ)
theorem
MeasureTheory.Measure.prod_dirac
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
[MeasureTheory.SFinite μ]
(y : β)
:
MeasureTheory.Measure.prod μ (MeasureTheory.Measure.dirac y) = MeasureTheory.Measure.map (fun (x : α) => (x, y)) μ
theorem
MeasureTheory.Measure.dirac_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
(x : α)
:
theorem
MeasureTheory.Measure.dirac_prod_dirac
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{x : α}
{y : β}
:
theorem
MeasureTheory.Measure.prod_add
{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]
(ν' : MeasureTheory.Measure β)
[MeasureTheory.SFinite ν']
:
MeasureTheory.Measure.prod μ (ν + ν') = MeasureTheory.Measure.prod μ ν + MeasureTheory.Measure.prod μ ν'
theorem
MeasureTheory.Measure.add_prod
{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]
(μ' : MeasureTheory.Measure α)
[MeasureTheory.SFinite μ']
:
MeasureTheory.Measure.prod (μ + μ') ν = MeasureTheory.Measure.prod μ ν + MeasureTheory.Measure.prod μ' ν
@[simp]
theorem
MeasureTheory.Measure.zero_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
(ν : MeasureTheory.Measure β)
:
MeasureTheory.Measure.prod 0 ν = 0
@[simp]
theorem
MeasureTheory.Measure.prod_zero
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
(μ : MeasureTheory.Measure α)
:
MeasureTheory.Measure.prod μ 0 = 0
theorem
MeasureTheory.Measure.map_prod_map
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{δ : Type u_7}
[MeasurableSpace δ]
{f : α → β}
{g : γ → δ}
(μa : MeasureTheory.Measure α)
(μc : MeasureTheory.Measure γ)
[MeasureTheory.SFinite μa]
[MeasureTheory.SFinite μc]
(hf : Measurable f)
(hg : Measurable g)
:
MeasureTheory.Measure.prod (MeasureTheory.Measure.map f μa) (MeasureTheory.Measure.map g μc) = MeasureTheory.Measure.map (Prod.map f g) (MeasureTheory.Measure.prod μa μc)
theorem
MeasureTheory.MeasurePreserving.skew_product
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{δ : Type u_7}
[MeasurableSpace δ]
{μa : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{μc : MeasureTheory.Measure γ}
{μd : MeasureTheory.Measure δ}
[MeasureTheory.SFinite μa]
[MeasureTheory.SFinite μc]
{f : α → β}
(hf : MeasureTheory.MeasurePreserving f μa μb)
{g : α → γ → δ}
(hgm : Measurable (Function.uncurry g))
(hg : ∀ᵐ (x : α) ∂μa, MeasureTheory.Measure.map (g x) μc = μd)
:
MeasureTheory.MeasurePreserving (fun (p : α × γ) => (f p.1, g p.1 p.2)) (MeasureTheory.Measure.prod μa μc)
(MeasureTheory.Measure.prod μb μd)
theorem
MeasureTheory.MeasurePreserving.prod
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{δ : Type u_7}
[MeasurableSpace δ]
{μa : MeasureTheory.Measure α}
{μb : MeasureTheory.Measure β}
{μc : MeasureTheory.Measure γ}
{μd : MeasureTheory.Measure δ}
[MeasureTheory.SFinite μa]
[MeasureTheory.SFinite μc]
{f : α → β}
{g : γ → δ}
(hf : MeasureTheory.MeasurePreserving f μa μb)
(hg : MeasureTheory.MeasurePreserving g μc μd)
:
MeasureTheory.MeasurePreserving (Prod.map f g) (MeasureTheory.Measure.prod μa μc) (MeasureTheory.Measure.prod μb 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
MeasureTheory.QuasiMeasurePreserving.prod_of_right
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{f : α × β → γ}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
{τ : MeasureTheory.Measure γ}
(hf : Measurable f)
[MeasureTheory.SFinite ν]
(h2f : ∀ᵐ (x : α) ∂μ, MeasureTheory.Measure.QuasiMeasurePreserving (fun (y : β) => f (x, y)) ν τ)
:
theorem
MeasureTheory.QuasiMeasurePreserving.prod_of_left
{α : Type u_7}
{β : Type u_8}
{γ : Type u_9}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{f : α × β → γ}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
{τ : MeasureTheory.Measure γ}
(hf : Measurable f)
[MeasureTheory.SFinite μ]
[MeasureTheory.SFinite ν]
(h2f : ∀ᵐ (y : β) ∂ν, MeasureTheory.Measure.QuasiMeasurePreserving (fun (x : α) => f (x, y)) μ τ)
:
theorem
AEMeasurable.prod_swap
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite μ]
[MeasureTheory.SFinite ν]
{f : β × α → γ}
(hf : AEMeasurable f (MeasureTheory.Measure.prod ν μ))
:
AEMeasurable (fun (z : α × β) => f (Prod.swap z)) (MeasureTheory.Measure.prod μ ν)
theorem
AEMeasurable.fst
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{f : α → γ}
(hf : AEMeasurable f μ)
:
AEMeasurable (fun (z : α × β) => f z.1) (MeasureTheory.Measure.prod μ ν)
theorem
AEMeasurable.snd
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{f : β → γ}
(hf : AEMeasurable f ν)
:
AEMeasurable (fun (z : α × β) => f z.2) (MeasureTheory.Measure.prod μ ν)
The Lebesgue integral on a product #
theorem
MeasureTheory.lintegral_prod_swap
{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]
(f : α × β → ENNReal)
:
∫⁻ (z : β × α), f (Prod.swap z) ∂MeasureTheory.Measure.prod ν μ = ∫⁻ (z : α × β), f z ∂MeasureTheory.Measure.prod μ ν
theorem
MeasureTheory.lintegral_prod_of_measurable
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
(f : α × β → ENNReal)
:
Measurable f → ∫⁻ (z : α × β), f z ∂MeasureTheory.Measure.prod μ ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ
Tonelli's Theorem: For ℝ≥0∞-valued measurable functions on α × β,
the integral of f is equal to the iterated integral.
theorem
MeasureTheory.lintegral_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
(f : α × β → ENNReal)
(hf : AEMeasurable f (MeasureTheory.Measure.prod μ ν))
:
∫⁻ (z : α × β), f z ∂MeasureTheory.Measure.prod μ ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ
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
MeasureTheory.lintegral_prod_symm
{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]
(f : α × β → ENNReal)
(hf : AEMeasurable f (MeasureTheory.Measure.prod μ ν))
:
∫⁻ (z : α × β), f z ∂MeasureTheory.Measure.prod μ ν = ∫⁻ (y : β), ∫⁻ (x : α), f (x, y) ∂μ ∂ν
The symmetric version 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
MeasureTheory.lintegral_prod_symm'
{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]
(f : α × β → ENNReal)
(hf : Measurable f)
:
∫⁻ (z : α × β), f z ∂MeasureTheory.Measure.prod μ ν = ∫⁻ (y : β), ∫⁻ (x : α), f (x, y) ∂μ ∂ν
The symmetric version 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
MeasureTheory.lintegral_lintegral
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
⦃f : α → β → ENNReal⦄
(hf : AEMeasurable (Function.uncurry f) (MeasureTheory.Measure.prod μ ν))
:
∫⁻ (x : α), ∫⁻ (y : β), f x y ∂ν ∂μ = ∫⁻ (z : α × β), f z.1 z.2 ∂MeasureTheory.Measure.prod μ 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
MeasureTheory.lintegral_lintegral_symm
{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]
⦃f : α → β → ENNReal⦄
(hf : AEMeasurable (Function.uncurry f) (MeasureTheory.Measure.prod μ ν))
:
∫⁻ (x : α), ∫⁻ (y : β), f x y ∂ν ∂μ = ∫⁻ (z : β × α), f z.2 z.1 ∂MeasureTheory.Measure.prod ν 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
MeasureTheory.lintegral_lintegral_swap
{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]
⦃f : α → β → ENNReal⦄
(hf : AEMeasurable (Function.uncurry f) (MeasureTheory.Measure.prod μ ν))
:
∫⁻ (x : α), ∫⁻ (y : β), f x y ∂ν ∂μ = ∫⁻ (y : β), ∫⁻ (x : α), f x y ∂μ ∂ν
Change the order of Lebesgue integration.
theorem
MeasureTheory.lintegral_prod_mul
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
{f : α → ENNReal}
{g : β → ENNReal}
(hf : AEMeasurable f μ)
(hg : AEMeasurable g ν)
:
∫⁻ (z : α × β), f z.1 * g z.2 ∂MeasureTheory.Measure.prod μ ν = (∫⁻ (x : α), f x ∂μ) * ∫⁻ (y : β), g y ∂ν
Marginals of a measure defined on a product #
noncomputable def
MeasureTheory.Measure.fst
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
(ρ : MeasureTheory.Measure (α × β))
:
Marginal measure on α obtained from a measure ρ on α × β, defined by ρ.map Prod.fst.
Equations
- MeasureTheory.Measure.fst ρ = MeasureTheory.Measure.map Prod.fst ρ
Instances For
theorem
MeasureTheory.Measure.fst_apply
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ρ : MeasureTheory.Measure (α × β)}
{s : Set α}
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.fst ρ) s = ↑↑ρ (Prod.fst ⁻¹' s)
theorem
MeasureTheory.Measure.fst_univ
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ρ : MeasureTheory.Measure (α × β)}
:
↑↑(MeasureTheory.Measure.fst ρ) Set.univ = ↑↑ρ Set.univ
@[simp]
theorem
MeasureTheory.Measure.fst_zero
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
:
instance
MeasureTheory.Measure.fst.instIsFiniteMeasure
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ρ : MeasureTheory.Measure (α × β)}
[MeasureTheory.IsFiniteMeasure ρ]
:
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.fst.instIsProbabilityMeasure
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ρ : MeasureTheory.Measure (α × β)}
[MeasureTheory.IsProbabilityMeasure ρ]
:
Equations
- ⋯ = ⋯
@[simp]
theorem
MeasureTheory.Measure.fst_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
[MeasureTheory.IsProbabilityMeasure ν]
:
theorem
MeasureTheory.Measure.fst_map_prod_mk₀
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{X : α → β}
{Y : α → γ}
{μ : MeasureTheory.Measure α}
(hY : AEMeasurable Y μ)
:
MeasureTheory.Measure.fst (MeasureTheory.Measure.map (fun (a : α) => (X a, Y a)) μ) = MeasureTheory.Measure.map X μ
theorem
MeasureTheory.Measure.fst_map_prod_mk
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{X : α → β}
{Y : α → γ}
{μ : MeasureTheory.Measure α}
(hY : Measurable Y)
:
MeasureTheory.Measure.fst (MeasureTheory.Measure.map (fun (a : α) => (X a, Y a)) μ) = MeasureTheory.Measure.map X μ
noncomputable def
MeasureTheory.Measure.snd
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
(ρ : MeasureTheory.Measure (α × β))
:
Marginal measure on β obtained from a measure on ρ α × β, defined by ρ.map Prod.snd.
Equations
- MeasureTheory.Measure.snd ρ = MeasureTheory.Measure.map Prod.snd ρ
Instances For
theorem
MeasureTheory.Measure.snd_apply
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ρ : MeasureTheory.Measure (α × β)}
{s : Set β}
(hs : MeasurableSet s)
:
↑↑(MeasureTheory.Measure.snd ρ) s = ↑↑ρ (Prod.snd ⁻¹' s)
theorem
MeasureTheory.Measure.snd_univ
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ρ : MeasureTheory.Measure (α × β)}
:
↑↑(MeasureTheory.Measure.snd ρ) Set.univ = ↑↑ρ Set.univ
@[simp]
theorem
MeasureTheory.Measure.snd_zero
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
:
instance
MeasureTheory.Measure.snd.instIsFiniteMeasure
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ρ : MeasureTheory.Measure (α × β)}
[MeasureTheory.IsFiniteMeasure ρ]
:
Equations
- ⋯ = ⋯
instance
MeasureTheory.Measure.snd.instIsProbabilityMeasure
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{ρ : MeasureTheory.Measure (α × β)}
[MeasureTheory.IsProbabilityMeasure ρ]
:
Equations
- ⋯ = ⋯
@[simp]
theorem
MeasureTheory.Measure.snd_prod
{α : Type u_1}
{β : Type u_3}
[MeasurableSpace α]
[MeasurableSpace β]
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure β}
[MeasureTheory.SFinite ν]
[MeasureTheory.IsProbabilityMeasure μ]
:
theorem
MeasureTheory.Measure.snd_map_prod_mk₀
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{X : α → β}
{Y : α → γ}
{μ : MeasureTheory.Measure α}
(hX : AEMeasurable X μ)
:
MeasureTheory.Measure.snd (MeasureTheory.Measure.map (fun (a : α) => (X a, Y a)) μ) = MeasureTheory.Measure.map Y μ
theorem
MeasureTheory.Measure.snd_map_prod_mk
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
[MeasurableSpace α]
[MeasurableSpace β]
[MeasurableSpace γ]
{X : α → β}
{Y : α → γ}
{μ : MeasureTheory.Measure α}
(hX : Measurable X)
:
MeasureTheory.Measure.snd (MeasureTheory.Measure.map (fun (a : α) => (X a, Y a)) μ) = MeasureTheory.Measure.map Y μ