Gaussian Smoothing Preserves the Mean

Proved

FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.smoothedLaw_mean_readable

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

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This statement identifies the mean of a Gaussian-smoothed probability law.

Let $E$ be a finite-dimensional complete real inner-product space with its Borel measurable structure. Let $μ$ be a probability measure on $E$ with MemLp id 2 μ, meaning that the identity map has finite second moment. For $s>0$ , let smoothedLaw μ s be the convolution with gaussianLaw s. Then

\int_E y\,d(\operatorname{smoothedLaw}(μ,s))(y) = \int_E y\,dμ(y).

This moment identity is useful when tracking centering through Gaussian regularization.

Original Lean declaration header

Within namespace FunctionalInequalities.KLS.LogConcave.GaussianSmoothing. Shown in its original module context. Proof omitted.

theorem smoothedLaw_mean (μ : Measure E) [IsProbabilityMeasure μ]
    (hμ2 : MemLp id 2 μ) {s : ℝ} (hs : 0 < s) :
    (∫ y : E, y ∂(smoothedLaw μ s)) = ∫ y : E, y ∂μ

Module navigation

Module: KLS/LogConcave/Density/GaussianSmoothing/Moments.

Definition modules: KLS/LogConcave/Density/GaussianSmoothing/Basic.

Supporting theorems: gaussianLaw_exp_norm_integrable, gaussianLaw_mean_zero.

Previous: gaussianLaw_inner_sq · Next: gaussianLaw_variance.

Source excerpt, lines 221–223 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_FunctionalInequalities_KLS_LogConcave_Density_GaussianSmoothing_Moments

open MeasureTheory ProbabilityTheory Filter Topology
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [CompleteSpace E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

Formal statement


theorem FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.smoothedLaw_mean_readable (μ : Measure E) [IsProbabilityMeasure μ]
    (hμ2 : MemLp id 2 μ) {s : ℝ} (hs : 0 < s) :
    (∫ y : E, y ∂(smoothedLaw μ s)) = ∫ y : E, y ∂μ := by sorry

Source

Lean source: https://github.com/human-ai-mathematics/kls-lean/blob/408915f/formalization/FunctionalInequalities/KLS/LogConcave/Density/GaussianSmoothing/Moments.lean#L220-L235 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.KLS.LogConcave.Density.GaussianSmoothing.Moments`, declaration `FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.smoothedLaw_mean`; the repository is private at upload time).

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