Almost-Everywhere Pathwise Equality from Right Continuity

Proved

LevyStochCalc.Ito.Picard.ae_all_eq_of_rightCont_readable

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

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

This promotes equality at each fixed time to simultaneous almost-everywhere equality on a time window.

Let $A,B:\mathbb R\to\Omega\to\mathbb R^n$ and $T\in\mathbb R$ . Assume: (1) for almost every $\omega$ , $t\mapsto A(t,\omega)$ is right-continuous; (2) for almost every $\omega$ , $t\mapsto B(t,\omega)$ is right-continuous; and (3) for every $t\in[0,T]$ , $A(t,\omega)=B(t,\omega)$ for almost every $\omega$ .

Then

\forall^{\mathrm{a.e.}}\omega,\ \forall t\in[0,T],\quad A(t,\omega)=B(t,\omega).

No probability or measurability structure beyond that appearing in the type is required. This is a standard separability upgrade using right continuity.

Original Lean declaration header

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

theorem ae_all_eq_of_rightCont {A B : ℝ → Ω → (Fin n → ℝ)} {T : ℝ}
    (hA : ∀ᵐ ω ∂P, ∀ t : ℝ,
      Filter.Tendsto (fun s => A s ω) (nhdsWithin t (Set.Ioi t)) (nhds (A t ω)))
    (hB : ∀ᵐ ω ∂P, ∀ t : ℝ,
      Filter.Tendsto (fun s => B s ω) (nhdsWithin t (Set.Ioi t)) (nhds (B t ω)))
    (h : ∀ t ∈ Set.Icc (0 : ℝ) T, ∀ᵐ ω ∂P, A t ω = B t ω) :
    ∀ᵐ ω ∂P, ∀ t ∈ Set.Icc (0 : ℝ) T, A t ω = B t ω

Module navigation

Module: LevyStochCalc/Ito/PicardGlobal.

Next: progressivelyMeasurable_globalPatch.

Source excerpt, lines 32–38 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Ito_PicardGlobal

/-
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/PicardGlobal.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory LevyStochCalc.Probability
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] [IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Ito.Picard.ae_all_eq_of_rightCont_readable {A B : ℝ → Ω → (Fin n → ℝ)} {T : ℝ}
    (hA : ∀ᵐ ω ∂P, ∀ t : ℝ,
      Filter.Tendsto (fun s => A s ω) (nhdsWithin t (Set.Ioi t)) (nhds (A t ω)))
    (hB : ∀ᵐ ω ∂P, ∀ t : ℝ,
      Filter.Tendsto (fun s => B s ω) (nhdsWithin t (Set.Ioi t)) (nhds (B t ω)))
    (h : ∀ t ∈ Set.Icc (0 : ℝ) T, ∀ᵐ ω ∂P, A t ω = B t ω) :
    ∀ᵐ ω ∂P, ∀ t ∈ Set.Icc (0 : ℝ) T, A t ω = B t ω := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Ito/PicardGlobal.lean#L29-L82 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Ito.PicardGlobal`, declaration `LevyStochCalc.Ito.Picard.ae_all_eq_of_rightCont`; 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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