Common Almost-Everywhere Subsequence for Finite Families

Proved

LevyStochCalc.Brownian.Ito.exists_seq_ae_tendsto_of_tendsto_lintegral_readable

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

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

Let $(\alpha,\mu)$ be a measurable measure space and let $\iota$ be a finite type. Suppose $u_i(c,\cdot)$ and $v(c,\cdot)$ are measurable real-valued functions, and for every $c\in\iota$ ,

\int^\infty \|u_i(c,a)-v(c,a)\|^2\,d\mu(a)\longrightarrow0.

Then there exists a subsequence map $m:\mathbb N\to\mathbb N$ with $i\le m(i)$ for every $i$ , such that for almost every $a$ , simultaneously for every $c\in\iota$ ,

u_{m(i)}(c,a)\longrightarrow v(c,a).

The squared norm is represented using ENNReal lower integrals.

Original Lean declaration header

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

theorem exists_seq_ae_tendsto_of_tendsto_lintegral {α : Type*} [MeasurableSpace α]
    {μ : Measure α} {ι : Type*} [Finite ι] {u : ℕ → ι → α → ℝ} {v : ι → α → ℝ}
    (hu : ∀ (i : ℕ) (c : ι), Measurable (u i c)) (hv : ∀ c : ι, Measurable (v c))
    (h : ∀ c : ι, Filter.Tendsto (fun i : ℕ => ∫⁻ a, (‖u i c a - v c a‖₊ : ℝ≥0∞) ^ 2 ∂μ)
      Filter.atTop (𝓝 0)) :
    ∃ ms : ℕ → ℕ, (∀ i : ℕ, i ≤ ms i) ∧
      ∀ᵐ a ∂μ, ∀ c : ι, Filter.Tendsto (fun i : ℕ => u (ms i) c a) Filter.atTop (𝓝 (v c a))

Module navigation

Module: LevyStochCalc/Ito/ItoFormulaUnboundedClamp.

Supporting theorems: tendsto_of_tendsto_sq_enorm.

Previous: tendsto_of_tendsto_sq_enorm · Next: ae_ae_tendsto_version_clamp.

Source excerpt, lines 397–403 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Ito_ItoFormulaUnboundedClamp

/-
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/ItoFormulaUnboundedClamp.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory LevyStochCalc.Brownian.Multidim
open scoped NNReal ENNReal Topology
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {n d : ℕ} {ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›}
omit [IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.Ito.exists_seq_ae_tendsto_of_tendsto_lintegral_readable {α : Type*} [MeasurableSpace α]
    {μ : Measure α} {ι : Type*} [Finite ι] {u : ℕ → ι → α → ℝ} {v : ι → α → ℝ}
    (hu : ∀ (i : ℕ) (c : ι), Measurable (u i c)) (hv : ∀ c : ι, Measurable (v c))
    (h : ∀ c : ι, Filter.Tendsto (fun i : ℕ => ∫⁻ a, (‖u i c a - v c a‖₊ : ℝ≥0∞) ^ 2 ∂μ)
      Filter.atTop (𝓝 0)) :
    ∃ ms : ℕ → ℕ, (∀ i : ℕ, i ≤ ms i) ∧
      ∀ᵐ a ∂μ, ∀ c : ι, Filter.Tendsto (fun i : ℕ => u (ms i) c a) Filter.atTop (𝓝 (v c a)) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Ito/ItoFormulaUnboundedClamp.lean#L394-L426 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Ito.ItoFormulaUnboundedClamp`, declaration `LevyStochCalc.Brownian.Ito.exists_seq_ae_tendsto_of_tendsto_lintegral`; 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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