Finite $S^2$ Control Implies Finite Bielecki Norm

Proved

LevyStochCalc.Ito.Picard.bieleckiNorm_lt_top_of_supL2_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concavevendored-levystochcalc

This shows that a finite windowwise $S^2$ bound gives a finite Bielecki norm at weight zero.

Let $P$ be a probability measure, let $T>0$ , and let $Z:\mathbb R\to\Omega\to\mathbb R^n$ . Assume that for every $T'>0$ ,

\int_\Omega\sup_{0\le t\le T'}\sum_{i<n}|Z(t,\omega,i)|^2\,dP(\omega)<\top.

Then

\operatorname{bieleckiNorm}(0,T,Z)<\top.

The supremum and integral are represented using ENNReal, and finiteness means strict inequality with $\top$ . This provides the integrability input needed for Picard-space arguments.

Original Lean declaration header

Within namespace LevyStochCalc.Ito.Picard. Shown in its original module context. Proof omitted.

theorem bieleckiNorm_lt_top_of_supL2 {T : ℝ} (hT : 0 < T) {Z : ℝ → Ω → (Fin n → ℝ)}
    (hS : ∀ T' : ℝ, 0 < T' →
      ∫⁻ ω, (⨆ t : Set.Icc (0 : ℝ) T', ∑ i, (‖Z (t : ℝ) ω i‖₊ : ℝ≥0∞) ^ 2) ∂P < ⊤) :
    bieleckiNorm (P := P) 0 T Z < ⊤

Module navigation

Module: LevyStochCalc/Ito/PicardWindow.

Definition modules: LevyStochCalc/Ito/PicardBieleckiNorm.

Next: bieleckiNorm_sub_lt_top_of_lt_top.

Source excerpt, lines 84–87 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Ito_PicardWindow

/-
Copyright (c) 2026 Christian Garry. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Garry
-/
-- Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (upstream path LevyStochCalc/Ito/PicardWindow.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open scoped NNReal ENNReal
universe u v
variable {Ω : Type u} [MeasurableSpace Ω] {E : Type v} [MeasurableSpace E]
variable {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {n d : ℕ}
omit [MeasurableSpace E]

Formal statement


theorem LevyStochCalc.Ito.Picard.bieleckiNorm_lt_top_of_supL2_readable {T : ℝ} (hT : 0 < T) {Z : ℝ → Ω → (Fin n → ℝ)}
    (hS : ∀ T' : ℝ, 0 < T' →
      ∫⁻ ω, (⨆ t : Set.Icc (0 : ℝ) T', ∑ i, (‖Z (t : ℝ) ω i‖₊ : ℝ≥0∞) ^ 2) ∂P < ⊤) :
    bieleckiNorm (P := P) 0 T Z < ⊤ := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Ito/PicardWindow.lean#L82-L103 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Ito.PicardWindow`, declaration `LevyStochCalc.Ito.Picard.bieleckiNorm_lt_top_of_supL2`; the repository is private at upload time). Vendored from https://github.com/cgarryZA/LevyStochCalc at commit 984dbf0b790031eac05d56fe5a84b65e15e1e932 (Apache-2.0; upstream copyright notice retained in the Lean text).

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Finite S2S^2S2 Control Implies Finite Bielecki Norm

Finite $S^2$ Control Implies Finite Bielecki Norm