Finite Bielecki Norm of a Difference

Proved

LevyStochCalc.Ito.Picard.bieleckiNorm_sub_lt_top_of_lt_top_readable

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

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

This proves finiteness of the Bielecki norm of a difference from finiteness of the two component norms.

Let $P$ be a probability measure, let $\beta\ge0$ , let $T\in\mathbb R$ , and let $X,Y:\mathbb R\to\Omega\to\mathbb R^n$ be jointly measurable. Assume

\operatorname{bieleckiNorm}(0,T,X)<\top,\qquad \operatorname{bieleckiNorm}(0,T,Y)<\top.

Then

\operatorname{bieleckiNorm}\bigl(\beta,T,(t,\omega,i)\mapsto X(t,\omega,i)-Y(t,\omega,i)\bigr)<\top.

The conclusion concerns the ENNReal-valued Bielecki norm and holds for every nonnegative weight $\beta$ . It is a basic closure property of the Picard space.

Original Lean declaration header

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

theorem bieleckiNorm_sub_lt_top_of_lt_top {β : ℝ} (hβ : 0 ≤ β) (T : ℝ)
    {X Y : ℝ → Ω → (Fin n → ℝ)}
    (hXm : Measurable (Function.uncurry X)) (hYm : Measurable (Function.uncurry Y))
    (hX : bieleckiNorm (P := P) 0 T X < ⊤) (hY : bieleckiNorm (P := P) 0 T Y < ⊤) :
    bieleckiNorm (P := P) β T (fun t ω i => X t ω i - Y t ω i) < ⊤

Module navigation

Module: LevyStochCalc/Ito/PicardWindow.

Definition modules: LevyStochCalc/Ito/PicardBieleckiNorm.

Previous: bieleckiNorm_lt_top_of_supL2 · Next: ae_eq_of_solvesOn.

Source excerpt, lines 107–111 · 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)
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_sub_lt_top_of_lt_top_readable {β : ℝ} (hβ : 0 ≤ β) (T : ℝ)
    {X Y : ℝ → Ω → (Fin n → ℝ)}
    (hXm : Measurable (Function.uncurry X)) (hYm : Measurable (Function.uncurry Y))
    (hX : bieleckiNorm (P := P) 0 T X < ⊤) (hY : bieleckiNorm (P := P) 0 T Y < ⊤) :
    bieleckiNorm (P := P) β T (fun t ω i => X t ω i - Y t ω i) < ⊤ := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Ito/PicardWindow.lean#L105-L131 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Ito.PicardWindow`, declaration `LevyStochCalc.Ito.Picard.bieleckiNorm_sub_lt_top_of_lt_top`; 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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