Integrating the conditional average recovers the joint integral

AutoSamplingTheory.condDistribIntegralMapIntegral · theorem · Teaching coverage

Statement

In the finite-measure conditional setting below, assume X and Y are μ-a.e. measurable and f is integrable under their joint image measure λ. Then integrating the conditional integral C(x) against the conditioning law m=X#μ equals integrating f(Xω,Yω) against μ. This is finite-measure disintegration, without probability normalization.

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).
• X and Y are μ-a.e. measurable, and f is λ-integrable.

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 condDistribIntegralMapIntegral {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}
    (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
    (hf : Integrable f (μ.map fun a => (X a, Y a))) :
    (∫ x, ∫ y, f (x, y) ∂ProbabilityTheory.condDistrib Y X μ x
        ∂μ.map X) =
      ∫ a, f (X a, Y a) ∂μ

theorem condDistribIntegralMapIntegral {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}
    (hX : AEMeasurable X μ) (hY : AEMeasurable Y μ)
    (hf : Integrable f (μ.map fun a => (X a, Y a))) :
    (∫ x, ∫ y, f (x, y) ∂ProbabilityTheory.condDistrib Y X μ x
        ∂μ.map X) =
      ∫ a, f (X a, Y a) ∂μ := by
  have hcomp := ProbabilityTheory.compProd_map_condDistrib
    (μ := μ) (X := X) (Y := Y) (mβ := inferInstance) hY
  have hfComp :
      Integrable f ((μ.map X) ⊗ₘ ProbabilityTheory.condDistrib Y X μ) := by
    rw [hcomp]
    exact hf
  calc
    (∫ x, ∫ y, f (x, y) ∂ProbabilityTheory.condDistrib Y X μ x
        ∂μ.map X)
        = ∫ z, f z ∂((μ.map X) ⊗ₘ
            ProbabilityTheory.condDistrib Y X μ) := by
          exact (Measure.integral_compProd
            (μ := μ.map X) (κ := ProbabilityTheory.condDistrib Y X μ)
            hfComp).symm
    _ = ∫ a, f (X a, Y a) ∂μ := by
          rw [hcomp]
          rw [integral_map (hX.prodMk hY) hf.1]

/-- Named-law variant of `condDistribIntegralMapIntegral`.

This is the paper-oriented form for `\hat\rho_s = Law(\hat X_s)`.
-/

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.
• For probability μ this is a two-stage expectation identity. For general finite μ, q(x) has mass one but the outer measure m has total mass μ(Ω); the reconstruction preserves that mass.
• No integration by parts, admissible log-ratio test or Fokker–Planck theorem.