Gaussian Law Has Zero Mean

Proved

FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.gaussianLaw_mean_zero_readable

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

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This is the mean-zero identity for gaussianLaw.

Let $E$ be a complete finite-dimensional real inner-product space with Borel measurable structure, let $s>0$ , and let $\gamma_s=\texttt{gaussianLaw }s$ . Then

\int_E y\,d\gamma_s(y)=0_E.

The conclusion is vector-valued and uses the Bochner integral. This is a basic symmetry property of the Gaussian noise and is used in the covariance and smoothing calculations.

Original Lean declaration header

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

theorem gaussianLaw_mean_zero {s : ℝ} (hs : 0 < s) :
    (∫ y : E, y ∂(gaussianLaw s)) = 0

Module navigation

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

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

Supporting theorems: gaussianLaw_exp_norm_integrable, gaussianLaw_laplace.

Previous: gaussianLaw_laplace · Next: gaussianLaw_inner_sq.

Source excerpt, lines 137–138 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_FunctionalInequalities_KLS_LogConcave_Density_GaussianSmoothing_Moments

variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [CompleteSpace E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

Formal statement


theorem FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.gaussianLaw_mean_zero_readable {s : ℝ} (hs : 0 < s) :
    (∫ y : E, y ∂(gaussianLaw s)) = 0 := by sorry

Source

Lean source: https://github.com/human-ai-mathematics/kls-lean/blob/408915f/formalization/FunctionalInequalities/KLS/LogConcave/Density/GaussianSmoothing/Moments.lean#L136-L162 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.KLS.LogConcave.Density.GaussianSmoothing.Moments`, declaration `FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.gaussianLaw_mean_zero`; the repository is private at upload time).

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