tendsto_setIntegral_norm_norm_ge_comp_toLp

Plain-English statement

- 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.

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 tendsto_setIntegral_norm_norm_ge_comp_toLp
    {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 in
        {x : Fin (n + 1) → ℝ |
          R ≤ ‖(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))‖},
        ‖H x‖ ∂μ)
      atTop (𝓝 0) := by
  let s : ℝ → Set (Fin (n + 1) → ℝ) :=
    fun R => {x | R ≤ ‖(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))‖}
  have hs_measurable : ∀ R, MeasurableSet (s R) := by
    intro R
    exact measurableSet_le measurable_const
      ((continuous_norm.comp (PiLp.continuous_toLp 2 _)).measurable)
  have hs_antitone : Antitone s := by
    intro R S hRS x hx
    exact hRS.trans hx
  have hs_iInter : ⋂ R : ℝ, s R = ∅ := by
    apply Set.Subset.antisymm
    · intro x hx
      have hx' := Set.mem_iInter.mp hx
        (‖(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))‖ + 1)
      simp only [s, Set.mem_ofPred_eq] at hx'
      linarith
    · exact Set.empty_subset _
  have htail := tendsto_setIntegral_of_antitone
    hs_measurable hs_antitone ⟨0, hH.norm.integrableOn⟩
  simpa [s, hs_iInter] using htail

/-- The trace contribution of `χ'.smulRight G` over the standard finite Pi
basis is exactly the scalar derivative `χ'` applied to `G`.

This is pure finite-dimensional linear algebra.  It identifies the cutoff
cross term used by the divergence product rule but proves no measurability,
integrability, convergence, or boundary result. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:365published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 cutoff route: make the L1 norm of any integrable raw-Pi field vanish outside expanding Euclidean balls

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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.
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.