integrable_expNeg_comp_toLp_of_lintegral_expNeg_ne_top

Plain-English statement

- Whole-space integrability of the unnormalized Gibbs weight in raw finite-Pi coordinates. The hypothesis is the finite `ℝ≥0∞` Gibbs mass on Euclidean space. Continuity supplies measurability, and Mathlib's volume-preserving `WithLp.toLp 2` equivalence transports integrability to the coordinate representation used by the cutoff lemmas. No test function, generator, tail limit, IBP, or invariant law is asserted here.

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 integrable_expNeg_comp_toLp_of_lintegral_expNeg_ne_top
    {n : ℕ}
    {V : EuclideanSpace ℝ (Fin (n + 1)) → ℝ}
    (hV : Continuous V)
    (hZ : (∫⁻ y : EuclideanSpace ℝ (Fin (n + 1)),
      ENNReal.ofReal (Real.exp (-V y)) ∂volume) ≠ ∞) :
    Integrable
      (fun x : Fin (n + 1) → ℝ =>
        Real.exp (-V (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))))
      volume := by
  have hweight_euclidean :
      Integrable (fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
        Real.exp (-V y)) volume := by
    exact
      (lintegral_ofReal_ne_top_iff_integrable
        hV.neg.rexp.aestronglyMeasurable
        (Filter.Eventually.of_forall fun y => Real.exp_nonneg (-V y))).1 hZ
  rw [← (PiLp.volume_preserving_toLp (Fin (n + 1))).integrable_comp_emb
    (MeasurableEquiv.toLp 2 _).measurableEmbedding] at hweight_euclidean
  simpa [Function.comp_def] using hweight_euclidean

/-- The unnormalized Gibbs mass outside expanding Euclidean balls tends to
zero, in the raw finite-Pi coordinates used by the radial cutoff route.

This combines finite Gibbs mass with the generic `L¹` tail theorem.  It is a
Gibbs-tail convergence certificate only: it does not identify a cutoff-field
divergence, prove weighted integration by parts, supply generator/semigroup
domains, or establish stationarity or invariance. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:1104published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 Gibbs-tail route: transport finite unnormalized Gibbs mass to an Integrable raw-Pi scalar weight shared by cutoff and tail consumers

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`have` creates a named intermediate mathematical fact.
`rw` rewrites by an established identity.
`exact` closes the current goal with an already typed term.
`simpa` closes the goal after a controlled simplification of a typed result.

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.
A Gibbs expression is not automatically a probability law or an invariant law.
A formal generator display does not establish a closed operator domain.
A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.