isProbabilityMeasure_withDensity_normalized_gibbs

Plain-English statement

A Gibbs density with a finite, nonzero normalization constant defines a probability measure after reciprocal normalization.

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.

Mathematical statement

If Z = integral exp(-V) dmu satisfies Z != 0 and Z != infinity, then mu.withDensity(Z^{-1} exp(-V)) has total mass one.

Intuition

This theorem isolates the normalization algebra from the harder analytic question of proving that the partition function is finite and nonzero.

Conditions

The Gibbs lintegral is nonzero.
The Gibbs lintegral is finite.
The base measure and potential are exactly those used in the density.

Why these conditions cannot be dropped

If Z is zero, reciprocal normalization is invalid.
If Z is infinite, ENNReal inversion collapses and cannot produce a probability law.

Proof route

Apply the generic normalized-withDensity theorem.
Instantiate its density with gibbsDensityENNReal.
Keep coercivity and envelope proofs outside this leaf.

Lean interface notes

IsProbabilityMeasure is a typeclass-valued proposition about total mass.
The theorem is a reusable normalization wrapper, not an invariance theorem.
lintegral values live in ENNReal, so zero and infinity are explicit side conditions.

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Lean statement

theorem isProbabilityMeasure_withDensity_normalized_gibbs
    (μ : MeasureTheory.Measure α) (V : α → ℝ)
    (h0 : ∫⁻ x, gibbsDensityENNReal V x ∂μ ≠ 0)
    (hfin : ∫⁻ x, gibbsDensityENNReal V x ∂μ ≠ ∞) :
    IsProbabilityMeasure
      (μ.withDensity fun x =>
        (∫⁻ y, gibbsDensityENNReal V y ∂μ)⁻¹ * gibbsDensityENNReal V x) :=
  RadonNikodym.isProbabilityMeasure_withDensity_normalized_lintegral μ
    (gibbsDensityENNReal V) h0 hfin

/-- A nonzero base measure and a finite a.e. envelope are enough to normalize a
Gibbs density into a probability measure.  This is the reusable contract that
later coercivity/growth leaves should target. -/

Open AutoSamplingTheory/TechnicalLemmas/Measure/Gibbs.lean:124published source at 7bcd37294df1