condDistribAeEqCondExpKernelMap

Plain-English statement

A conditional distribution agrees almost everywhere with the conditional-expectation kernel mapped through the sampled random variable.

Mathematical statement

For measurable X and Y and measurable s, condDistrib(Y|X)(X(a),s) equals the mapped condExpKernel value for mu-almost every a.

Intuition

Mathlib exposes conditional laws through kernels and only up to almost-everywhere equality. This bridge aligns the orientation used by sampling proofs without choosing a pointwise version.

Conditions

X and Y are measurable.
The source and target standard-Borel hypotheses required by Mathlib are present.
The base measure is finite.
The queried set is measurable.

Why these conditions cannot be dropped

Conditional expectations and regular conditional distributions are version-dependent and generally only almost-everywhere defined.
Standard-Borel hypotheses support the regular conditional-distribution construction.
Kernel evaluation on a nonmeasurable set is outside the API contract.

Proof route

Use Mathlib's conditional-distribution versus mapped conditional-expectation-kernel theorem.
Swap the X/Y orientation to match the ASTIS consumer.
Preserve equality in the ae filter rather than strengthening it pointwise.

Lean learning studio · mathematics → formal proof

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Lean statement

theorem condDistribAeEqCondExpKernelMap {Ω β γ : Type*}
    [MeasurableSpace Ω] [MeasurableSpace β] [MeasurableSpace γ]
    [StandardBorelSpace Ω] [StandardBorelSpace γ] [Nonempty γ]
    {μ : Measure Ω} [IsFiniteMeasure μ] {X : Ω → β} {Y : Ω → γ}
    (hX : Measurable X) (hY : Measurable Y) {s : Set γ}
    (hs : MeasurableSet s) :
    (fun a => ProbabilityTheory.condDistrib Y X μ (X a) s) =ᵐ[μ]
      fun a =>
        (ProbabilityTheory.condExpKernel μ
            ((inferInstance : MeasurableSpace β).comap X)).map Y a s := by
  exact ProbabilityTheory.condDistrib_apply_ae_eq_condExpKernel_map
    (μ := μ) (X := Y) (Y := X) hY hX hs

/-- Sample-space component-version bridge from `condExpKernel.map` to
`condDistrib`.

For the SALD `condC` field, this isolates the remaining Mathlib-facing
boundary after `condDistrib` and `condExpKernel.map` have been aligned as
measure-valued kernels almost everywhere.  It turns a selected
`condExpKernel.map` version of the component field into the displayed
`condDistrib` integral equality after composing with `hat X_s`.  The theorem
does not prove the measure-valued kernel equality or choose the component
version; those remain the smaller conditional-kernel obligations.
-/

Lean interface notes

=ᵐ[mu] is Filter.Eventually equality in the almost-everywhere filter.
MeasurableSpace.comap X is the sigma-algebra generated by the conditioning variable.
Kernel.map Y pushes the conditional kernel through Y.