finite_klDiv_and_toReal_eq_integral_normalizedLogRatio

Plain-English statement

- If the explicit normalized log-ratio of two integrable exponential tilts is integrable under the left tilt, then their canonical KL divergence is finite and its real value is the integral of that explicit representative. The integrability premise is deliberately stated for the displayed representative. Almost-everywhere equality is used only to transfer integrability and the integral, never to transfer classical derivatives.

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 finite_klDiv_and_toReal_eq_integral_normalizedLogRatio
    (mu : Measure X) [SigmaFinite mu] [NeZero mu] (f g : X → ℝ)
    (hf : Measurable f)
    (hfi : Integrable (fun x => Real.exp (f x)) mu)
    (hgi : Integrable (fun x => Real.exp (g x)) mu)
    (hrep : Integrable
      (fun x =>
        f x - Real.log (∫ z, Real.exp (f z) ∂mu) - g x +
          Real.log (∫ z, Real.exp (g z) ∂mu))
      (mu.tilted f)) :
    InformationTheory.klDiv (mu.tilted f) (mu.tilted g) ≠ ⊤ ∧
      (InformationTheory.klDiv (mu.tilted f) (mu.tilted g)).toReal =
        ∫ x,
          (f x - Real.log (∫ z, Real.exp (f z) ∂mu) - g x +
            Real.log (∫ z, Real.exp (g z) ∂mu)) ∂(mu.tilted f) := by
  let _ : IsProbabilityMeasure (mu.tilted f) := isProbabilityMeasure_tilted hfi
  let _ : IsProbabilityMeasure (mu.tilted g) := isProbabilityMeasure_tilted hgi
  have hae := TiltedLogRatio.llr_tilted_tilted_ae mu f g hf hfi hgi
  have hac : mu.tilted f ≪ mu.tilted g :=
    (tilted_absolutelyContinuous mu f).trans (absolutelyContinuous_tilted hgi)
  have hllr : Integrable (MeasureTheory.llr (mu.tilted f) (mu.tilted g))
      (mu.tilted f) :=
    hrep.congr hae.symm
  constructor
  · exact InformationTheory.klDiv_ne_top hac hllr
  · rw [InformationTheory.toReal_klDiv_of_measure_eq hac (by simp)]
    exact integral_congr_ae hae

end

end AutoSamplingTheory.TechnicalLemmas.InformationTheory.TiltedKL

Open AutoSamplingTheory/TechnicalLemmas/InformationTheory/TiltedKL.lean:28published source at ed709a7f320d

Proof architecture

SPHMC Lemma 4.17 route: evaluate canonical KL through the explicit normalized tilt representative without differentiating the canonical llr

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Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

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.