integral_withDensity_lintegral_inv_mul_gibbsDensityENNReal_eq_integral_lintegral_inv_mul_exp_smul_of_neZero

Plain-English statement

- Nonzero-base-measure specialization of the source-facing Gibbs integral rewrite. It discharges the nonzero Gibbs normalizer from positivity of `exp (-V)` and `[NeZero μ]`. This remains only a Bochner-integral rewrite. If a later theorem needs a probability measure, it must also supply the finite-normalizer hypothesis to the separate Gibbs probability leaf.

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 integral_withDensity_lintegral_inv_mul_gibbsDensityENNReal_eq_integral_lintegral_inv_mul_exp_smul_of_neZero
    (μ : Measure α) [NeZero μ] {V : α → ℝ} (hV : AEMeasurable V μ) (g : α → F) :
    ∫ x, g x ∂μ.withDensity
        (fun x =>
          (∫⁻ y, Measure.Gibbs.gibbsDensityENNReal V y ∂μ)⁻¹ *
            Measure.Gibbs.gibbsDensityENNReal V x) =
      ∫ x,
        ((∫⁻ y, Measure.Gibbs.gibbsDensityENNReal V y ∂μ).toReal⁻¹ *
          Real.exp (-V x)) • g x ∂μ :=
  integral_withDensity_lintegral_inv_mul_gibbsDensityENNReal_eq_integral_lintegral_inv_mul_exp_smul
    μ hV (Measure.Gibbs.lintegral_gibbsDensityENNReal_ne_zero μ hV) g

end GibbsIntegral
end Measure
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/Measure/GibbsIntegral.lean:76published source at 7bcd37294df1

Proof architecture

Chewi MEAS/DENS/SDE root: use textbook-shaped `Z = ∫ exp(-V)` Gibbs density inside Bochner test-function integrals without separately passing the nonzero-normalizer proof

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Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
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.