Integrability From a Probability Tilt Representation

Proved

FunctionalInequalities.KLS.LogConcave.integrable_exp_neg_of_eq_tilted_readable

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

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This is an integrability consequence of a tilted probability representation.

Let $V$ be a measurable space, let $m$ be a measure on $V$ , let $W:V\to\mathbb R$ , and let $\nu$ be a probability measure. If

\nu=m.\operatorname{tilted}(-W),

then the unnormalized density is integrable:

x\longmapsto e^{-W(x)}\quad\text{is integrable with respect to }m.

No normed-space, finite-dimensional, or Borel assumptions are required by the type. The conclusion uses that a tilted measure identified with a probability measure cannot arise from a nonintegrable density.

Original Lean declaration header

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

theorem integrable_exp_neg_of_eq_tilted {m : Measure V} {W : V → ℝ}
    (hνW : (ν : Measure V) = m.tilted (-W)) :
    Integrable (fun x => Real.exp (-W x)) m

Module navigation

Module: KLS/LogConcave/Density/Tilted.

Next: integral_exp_neg_pos_of_eq_tilted.

Source excerpt, lines 30–32 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_FunctionalInequalities_KLS_LogConcave_Density_Tilted

open MeasureTheory Set
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V]
  [MeasurableSpace V] [BorelSpace V] {ν : MeasureTheory.ProbabilityMeasure V}
omit [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [BorelSpace V]

Formal statement


theorem FunctionalInequalities.KLS.LogConcave.integrable_exp_neg_of_eq_tilted_readable {m : Measure V} {W : V → ℝ}
    (hνW : (ν : Measure V) = m.tilted (-W)) :
    Integrable (fun x => Real.exp (-W x)) m := by sorry

Source

Lean source: https://github.com/human-ai-mathematics/kls-lean/blob/408915f/formalization/FunctionalInequalities/KLS/LogConcave/Density/Tilted.lean#L28-L45 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.KLS.LogConcave.Density.Tilted`, declaration `FunctionalInequalities.KLS.LogConcave.integrable_exp_neg_of_eq_tilted`; the repository is private at upload time).

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