Set Integral Factorization Under Independence

Proved

LevyStochCalc.Probability.setIntegral_eq_measure_mul_of_indep_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concavevendored-levystochcalc

Let $m\leq m_\Omega$ be measurable spaces on a probability space $(\Omega,P)$ . Let $X:\Omega\to\mathcal X$ be measurable, where $\mathcal X$ is a topological measurable Borel space, and assume the sigma-algebra $m$ is independent of the comap sigma-algebra generated by $X$ . If $A$ is measurable in $m$ and $f:\mathcal X\to\mathbb R$ is bounded continuous, then

\int_A f(X(\omega))\,dP(\omega)=(P(A)).\operatorname{toReal}\,\int_\Omega f(X(\omega))\,dP(\omega).

The conclusion uses the real coercion toReal of the probability $P(A)$ . This is a reusable factorization identity for independent events and bounded observables.

Original Lean declaration header

Within namespace LevyStochCalc.Probability. Shown in its original module context. Proof omitted.

lemma setIntegral_eq_measure_mul_of_indep (hm : m ≤ mΩ) {X : Ω → 𝓧}
    (hX : Measurable[mΩ] X) (hind : Indep m (MeasurableSpace.comap X inferInstance) P)
    {A : Set Ω} (hA : MeasurableSet[m] A) (f : 𝓧 →ᵇ ℝ) :
    ∫ ω in A, f (X ω) ∂P = (P A).toReal * ∫ ω, f (X ω) ∂P

Module navigation

Module: LevyStochCalc/Probability/IndepLimit.

Next: indep_comap_of_tendsto_ae.

Source excerpt, lines 29–32 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Probability_IndepLimit

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Probability/IndepLimit.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory Filter Topology
open scoped BoundedContinuousFunction
variable {Ω 𝓧 : Type*} {m mΩ : MeasurableSpace Ω} {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  [TopologicalSpace 𝓧] [MeasurableSpace 𝓧] [BorelSpace 𝓧] [HasOuterApproxClosed 𝓧]
omit [HasOuterApproxClosed 𝓧]

Formal statement


lemma LevyStochCalc.Probability.setIntegral_eq_measure_mul_of_indep_readable (hm : m ≤ mΩ) {X : Ω → 𝓧}
    (hX : Measurable[mΩ] X) (hind : Indep m (MeasurableSpace.comap X inferInstance) P)
    {A : Set Ω} (hA : MeasurableSet[m] A) (f : 𝓧 →ᵇ ℝ) :
    ∫ ω in A, f (X ω) ∂P = (P A).toReal * ∫ ω, f (X ω) ∂P := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Probability/IndepLimit.lean#L27-L52 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Probability.IndepLimit`, declaration `LevyStochCalc.Probability.setIntegral_eq_measure_mul_of_indep`; the repository is private at upload time). Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (Apache-2.0; upstream copyright notice retained in the Lean text).

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