The conditional integral is integrable under the conditioning law

AutoSamplingTheory.condDistribIntegralMapIntegrable · theorem · Teaching coverage

Statement

In the finite-measure conditional-distribution setting and notation below, assume Y is μ-a.e. measurable. If f is integrable under the joint image measure λ, then C is integrable under m. No a.e. measurability assumption on X is required by this law-space adapter.

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).
• Y is μ-a.e. measurable.
• f is integrable under λ.

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 condDistribIntegralMapIntegrable {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] {X : Ω → β} {Y : Ω → γ}
    {f : β × γ → F}
    (hY : AEMeasurable Y μ)
    (hf : Integrable f (μ.map fun a => (X a, Y a))) :
    Integrable
      (fun x => ∫ y, f (x, y) ∂ProbabilityTheory.condDistrib Y X μ x)
      (μ.map X)

theorem condDistribIntegralMapIntegrable {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] {X : Ω → β} {Y : Ω → γ}
    {f : β × γ → F}
    (hY : AEMeasurable Y μ)
    (hf : Integrable f (μ.map fun a => (X a, Y a))) :
    Integrable
      (fun x => ∫ y, f (x, y) ∂ProbabilityTheory.condDistrib Y X μ x)
      (μ.map X) := by
  exact hf.integral_condDistrib_map hY

/-- Disintegrate an integral through the `condDistrib` kernel.

For the SALD conditional drift in `appendix.tex:1368-1377`, instantiate `X`
with `hat X_s`, `Y` with `X_k^eta`, and `f` with the weak test-gradient
pairing against one frozen component.  This proves the map-law
conditional-integral identity behind the canonical `condDistrib` component
generator action; it does not prove the weak Fokker--Planck equation,
boundary integration by parts, or log-ratio admissibility.
-/

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.
• X need not be a.e. measurable in this exact signature; without it, the displayed image measures retain Mathlib's totalized-map semantics.
• No specified SALD component version, path law, or weak PDE is constructed.