Plain-English statement
- Multiplication by the PiLp-wrapped radial cutoff converges to the identity under integration for every integrable real normed-space-valued source field. The statement is measure-generic and uses only integrability of the source. It proves the cutoff main-term limit, but no Gibbs-specific integrability, cutoff-gradient estimate, integration by parts, generator-domain result, or invariant-law statement.
Read the mathematics first, then descend into Lean
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Proof architecture
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How to read the exact Lean declaration
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- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
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Syntax used on this page
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Lean statement
theorem tendsto_integral_radialSmoothCutoff_comp_toLp_smul
{n : ℕ} {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
{μ : Measure (Fin (n + 1) → ℝ)}
{H : (Fin (n + 1) → ℝ) → F}
(hH : Integrable H μ) :
Tendsto
(fun R : ℝ => ∫ x,
Cutoff.radialSmoothCutoff R
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) • H x ∂μ)
atTop (𝓝 (∫ x, H x ∂μ)) := by
have hmeas :
∀ᶠ R : ℝ in atTop,
AEStronglyMeasurable
(fun x : Fin (n + 1) → ℝ =>
Cutoff.radialSmoothCutoff R
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) • H x) μ := by
filter_upwards [eventually_gt_atTop (0 : ℝ)] with R hR
exact
(((Cutoff.radialSmoothCutoff_contDiff hR).continuous.comp
(PiLp.continuous_toLp 2 _)).aestronglyMeasurable).smul
hH.aestronglyMeasurable
have hdom :
∀ᶠ R : ℝ in atTop, ∀ᵐ x ∂μ,
‖Cutoff.radialSmoothCutoff R
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) • H x‖ ≤
‖H x‖ := by
filter_upwards with R
filter_upwards with x
have hcutoff :=
Cutoff.radialSmoothCutoff_mem_Icc R
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))
rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg hcutoff.1]
exact mul_le_of_le_one_left (norm_nonneg _) hcutoff.2
have hpoint :
∀ᵐ x ∂μ,
Tendsto
(fun R : ℝ =>
Cutoff.radialSmoothCutoff R
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) • H x)
atTop (𝓝 (H x)) := by
filter_upwards with x
simpa using
(Cutoff.radialSmoothCutoff_tendsto_one
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))).smul_const (H x)
exact MeasureTheory.tendsto_integral_filter_of_dominated_convergence
(μ := μ) (l := atTop)
(F := fun R x =>
Cutoff.radialSmoothCutoff R
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) • H x)
(f := H) (fun x => ‖H x‖) hmeas hdom hH.norm hpoint
/-- The `L¹` norm of an integrable field on the complement of an expanding
Euclidean ball tends to zero, expressed in raw finite-Pi coordinates.
The tail sets are `R ≤ ‖WithLp.toLp 2 x‖`. They form an antitone family with
empty intersection, so Mathlib's antitone set-integral convergence theorem
applies to `‖H‖`. This is measure-generic and contains no Gibbs, generator,
integration-by-parts, or invariance semantics. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:307published source at 7bcd37294df1
Proof architecture
Chewi Ch.1 cutoff-smul route: pass the main integrable field through the PiLp-wrapped radial cutoff by dominated convergence
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.
- `filter_upwards` moves an almost-everywhere or eventual statement into a pointwise local context.
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.