A chosen a.e. version inherits conditional-integral regularity

AutoSamplingTheory.condDistribIntegralNamedFieldRegularity · theorem · Teaching coverage

Statement

In the conditional setting below, let ρ̂=X#μ and v:β→F be a chosen field equal to C ρ̂-almost everywhere. If Y is μ-a.e. measurable and f is both a.e. strongly measurable and integrable under λ, then v is a.e. strongly measurable and integrable under ρ̂.

All objects and hypotheses

• Ω, β and γ are measurable spaces; γ is Standard Borel and nonempty.
• μ is a finite measure on Ω, not necessarily a probability; X:Ω→β and Y:Ω→γ are functions.
• F is a real normed vector space (NormedAddCommGroup and NormedSpace ℝ); no CompleteSpace assumption is added. The integrand f:β×γ→F is as specified below.
• Notation: λ=(ω↦(Xω,Yω))#μ, m=X#μ, q(x)=condDistrib(Y|X;μ)(x), and C(x)=∫f(x,y)dq(x)(y).
• ρ̂ is a measure on β equal to X#μ, and v:β→F is a chosen field.
• Y is μ-a.e. measurable; hfMeas explicitly supplies AEStronglyMeasurable f λ and hfInt explicitly supplies Integrable f λ (both premises are retained).
• hfield supplies C=v ρ̂-a.e.; X a.e. measurability is not separately required.

Notation and interpretation

Pushforward law

P is an arbitrary measure unless explicitly declared finite or a probability. The word law here abbreviates a pushforward measure; probability-language interpretations require normalization separately. Mathlib totalizes map to zero when X is not a.e. measurable; unqualified map-congruence and some law-space adapters intentionally retain that generality.

\[(X_\#\mu)(A)=\mu(X^{-1}A)\quad\text{when the measurable pushforward interpretation applies}\]

A.e. and strong measurability

A.e. means outside a μ-null set. AEMeasurable means equality a.e. to a measurable map; AEStronglyMeasurable (abbreviated AESM) means equality a.e. to a strongly measurable function, which is approximable by simple functions.

\[f=g\quad\mu\text{-a.e.}\]

Integrability and Bochner integrals

L¹ in the teaching formulas means Integrable, not a newly defined quotient-space element. ∫ denotes Mathlib's totalized Bochner integral: it is zero for nonintegrable functions, and also for a codomain lacking completeness. Do not infer integrability or a genuine finite expectation from an unqualified integral equality. Real-valued integrals have a complete codomain; missing integrability still matters.

\[f\in L^1(\mu)\quad\Longleftrightarrow\quad f\text{ is a.e. strongly measurable and }\int^{\!-}\|f\|\,d\mu<\infty\]

Conditional integral notation

The selected conditional distribution is a probability kernel, characterized as a conditional law only almost everywhere under the conditioning marginal. No pointwise choice, simultaneous equality on all events from a single-event a.e. theorem, or null-fiber support is asserted. Finite non-probability measures are reconstructed with their original total mass; any version changes on marginal-null inputs must retain a measurable kernel.

\[\lambda=(X,Y)_\#\mu,\quad m=X_\#\mu,\quad q(x)=\operatorname{condDistrib}(Y\mid X;\mu)(x),\quad C(x)=\int f(x,y)\,dq(x)(y)\]

Mathematical proof

theorem condDistribIntegralNamedFieldRegularity {Ω β γ F : Type*}
    [MeasurableSpace Ω] [MeasurableSpace β] [MeasurableSpace γ]
    [NormedAddCommGroup F] [NormedSpace ℝ F]
    [StandardBorelSpace γ] [Nonempty γ]
    {μ : Measure Ω} [IsFiniteMeasure μ] {hatRho : Measure β}
    {X : Ω → β} {Y : Ω → γ} {f : β × γ → F} {field : β → F}
    (hhatRho : hatRho = μ.map X)
    (hY : AEMeasurable Y μ)
    (hfMeas : AEStronglyMeasurable f (μ.map fun a => (X a, Y a)))
    (hfInt : Integrable f (μ.map fun a => (X a, Y a)))
    (hfield :
      (fun x => ∫ y, f (x, y) ∂ProbabilityTheory.condDistrib Y X μ x)
        =ᵐ[hatRho] field) :
    AEStronglyMeasurable field hatRho ∧ Integrable field hatRho

theorem condDistribIntegralNamedFieldRegularity {Ω β γ F : Type*}
    [MeasurableSpace Ω] [MeasurableSpace β] [MeasurableSpace γ]
    [NormedAddCommGroup F] [NormedSpace ℝ F]
    [StandardBorelSpace γ] [Nonempty γ]
    {μ : Measure Ω} [IsFiniteMeasure μ] {hatRho : Measure β}
    {X : Ω → β} {Y : Ω → γ} {f : β × γ → F} {field : β → F}
    (hhatRho : hatRho = μ.map X)
    (hY : AEMeasurable Y μ)
    (hfMeas : AEStronglyMeasurable f (μ.map fun a => (X a, Y a)))
    (hfInt : Integrable f (μ.map fun a => (X a, Y a)))
    (hfield :
      (fun x => ∫ y, f (x, y) ∂ProbabilityTheory.condDistrib Y X μ x)
        =ᵐ[hatRho] field) :
    AEStronglyMeasurable field hatRho ∧ Integrable field hatRho := by
  have hmeas :
      AEStronglyMeasurable
        (fun x => ∫ y, f (x, y) ∂ProbabilityTheory.condDistrib Y X μ x)
        hatRho :=
    condDistribIntegralNamedLawAEStronglyMeasurable hhatRho hY hfMeas
  have hint :
      Integrable
        (fun x => ∫ y, f (x, y) ∂ProbabilityTheory.condDistrib Y X μ x)
        hatRho :=
    condDistribIntegralNamedLawIntegrable hhatRho hY hfInt
  exact ⟨hmeas.congr hfield, hint.congr hfield⟩

/-- A named probability measure or time-indexed law in a paper proof. -/

Scope and omitted-condition boundaries

• The selected conditional distribution is a probability kernel, characterized as a conditional law only almost everywhere under the conditioning marginal. No pointwise choice, simultaneous equality on all events from a single-event a.e. theorem, or null-fiber support is asserted.
• ∫ denotes Mathlib's totalized Bochner integral: it is zero for nonintegrable functions, and also for a codomain lacking completeness. Do not infer integrability or a genuine finite expectation from an unqualified integral equality. Real-valued integrals have a complete codomain; missing integrability still matters.
• Does not choose v or prove its version equality. Both explicit regularity premises remain in the source signature. No whole-PDE regularity follows.