Gaussian Exponential Norm Integrability

Proved

FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.gaussianLaw_exp_norm_integrable_readable

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

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This is a moment estimate for the Gaussian law gaussianLaw.

Let $E$ be a finite-dimensional real inner-product space with Borel measurable structure, let $s\in\mathbb R$ satisfy $s>0$ , and let $\gamma_s$ denote gaussianLaw s. Then

\int_E e^{\|y\|}\,d\gamma_s(y)<\infty.

Equivalently, $y\mapsto e^{\|y\|}$ is integrable with respect to $\gamma_s$ . Completeness of $E$ is omitted from the declaration. This estimate supplies exponential integrability for Gaussian Laplace-transform and moment calculations.

Original Lean declaration header

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

theorem gaussianLaw_exp_norm_integrable {s : ℝ} (hs : 0 < s) :
    Integrable (fun y : E => Real.exp ‖y‖) (gaussianLaw s)

Module navigation

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

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

Next: gaussianLaw_laplace.

Source excerpt, lines 42–43 · 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]
omit [CompleteSpace E]

Formal statement


theorem FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.gaussianLaw_exp_norm_integrable_readable {s : ℝ} (hs : 0 < s) :
    Integrable (fun y : E => Real.exp ‖y‖) (gaussianLaw s) := by sorry

Source

Lean source: https://github.com/human-ai-mathematics/kls-lean/blob/408915f/formalization/FunctionalInequalities/KLS/LogConcave/Density/GaussianSmoothing/Moments.lean#L40-L57 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.KLS.LogConcave.Density.GaussianSmoothing.Moments`, declaration `FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.gaussianLaw_exp_norm_integrable`; the repository is private at upload time).

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