isProbabilityMeasure_withDensity_ofReal_exp_of_integral_eq_one

Plain-English statement

- A real exponential tilt with Bochner integral one defines a probability measure through `withDensity`. This is the small ASTIS-owned version of the exponential-tilt normalization pattern used in entropy-duality and Girsanov arguments.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem isProbabilityMeasure_withDensity_ofReal_exp_of_integral_eq_one
    (μ : MeasureTheory.Measure α) {U : α → ℝ}
    (hU_int : Integrable (fun x => Real.exp (U x)) μ)
    (hU_mass : ∫ x, Real.exp (U x) ∂μ = 1) :
    IsProbabilityMeasure
      (μ.withDensity fun x => ENNReal.ofReal (Real.exp (U x))) := by
  refine isProbabilityMeasure_withDensity_of_lintegral_eq_one μ
    (fun x => ENNReal.ofReal (Real.exp (U x))) ?_
  rw [← ofReal_integral_eq_lintegral_ofReal hU_int
    (ae_of_all _ fun _ => (Real.exp_pos _).le), hU_mass]
  norm_num

/-- A density with finite lintegral defines a finite measure after
`withDensity`. -/

Open AutoSamplingTheory/TechnicalLemmas/Measure/RadonNikodym.lean:142published source at 7bcd37294df1

Proof architecture

log-concave sampling DENS/MEAS/PATH root: normalize real exponential tilts for entropy duality, Gibbs variational formulas, and finite-dimensional Girsanov/RN routes

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Why the statement has this shape

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Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.
Almost-everywhere hypotheses depend on the stated measure and representative.
Density statements retain normalization and absolute-continuity prerequisites.
A Gibbs expression is not automatically a probability law or an invariant law.
A finite cylinder identity is not automatically a path-space change-of-measure theorem.