Compact-Set Haar Domination By A Tilted Probability Law

Proved

FunctionalInequalities.KLS.LogConcave.restrict_le_smul_of_eq_tilted_readable

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

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This theorem gives a finite measure domination on compact sets.

Let $V$ be a measurable Borel normed space, let $m$ be an additive Haar measure, let $\nu$ be a probability measure, and let $U:V\to\mathbb R$ be continuous with $\nu=m.\operatorname{tilted}(-U)$ . For every compact set $K\subseteq V$ , there exists $c\in[0,\infty]$ such that $c\ne\infty$ and

m\!\restriction_K\le c\,\nu.

Thus the restriction of Haar measure to any compact set is dominated by a finite scalar multiple of the tilted law. The result is used to obtain local integrability with respect to Haar measure from $L^2(\nu)$ information.

Original Lean declaration header

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

theorem restrict_le_smul_of_eq_tilted (hνU : (ν : Measure V) = m.tilted (-U)) (hU : Continuous U)
    {K : Set V} (hK : IsCompact K) :
    ∃ c : ENNReal, c ≠ ⊤ ∧ m.restrict K ≤ c • (ν : Measure V)

Module navigation

Module: KLS/LogConcave/WeightedGradient/Kernel.

Supporting theorems: integrable_exp_neg_of_eq_tilted.

Previous: integral_eq_integral_mul_exp · Next: tendsto_integral_mul_of_tendsto.

Source excerpt, lines 81–83 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_FunctionalInequalities_KLS_LogConcave_WeightedGradient_Kernel

open MeasureTheory Filter Topology InnerProductSpace
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
  [MeasurableSpace V] [BorelSpace V] [SecondCountableTopology V] [FiniteDimensional ℝ V]
  {ν : MeasureTheory.ProbabilityMeasure V} (m : MeasureTheory.Measure V) [m.IsAddHaarMeasure] {U : V → ℝ}
omit [InnerProductSpace ℝ V] [SecondCountableTopology V]
  [FiniteDimensional ℝ V]

Formal statement


theorem FunctionalInequalities.KLS.LogConcave.restrict_le_smul_of_eq_tilted_readable (hνU : (ν : Measure V) = m.tilted (-U)) (hU : Continuous U)
    {K : Set V} (hK : IsCompact K) :
    ∃ c : ENNReal, c ≠ ⊤ ∧ m.restrict K ≤ c • (ν : Measure V) := by sorry

Source

Lean source: https://github.com/human-ai-mathematics/kls-lean/blob/408915f/formalization/FunctionalInequalities/KLS/LogConcave/WeightedGradient/Kernel.lean#L77-L134 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.KLS.LogConcave.WeightedGradient.Kernel`, declaration `FunctionalInequalities.KLS.LogConcave.restrict_le_smul_of_eq_tilted`; the repository is private at upload time).

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