L¹ Convergence of Squares from L² Convergence

Proved

LevyStochCalc.Brownian.Ito.tendsto_eLpNorm_one_sq_sub_readable

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

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

Let $a_n:\Omega\to\mathbb R$ be a filter-indexed family and let $b:\Omega\to\mathbb R$ . Assume every $a_n$ and $b$ is almost-everywhere measurable, $\|b\|_{L^2(\mu)}\ne\infty$ , and

\|a_n-b\|_{L^2(\mu)}\longrightarrow 0

along a filter $l$ . Then

\left\|a_n^2-b^2\right\|_{L^1(\mu)}\longrightarrow 0

along $l$ . The norms are the extended $L^p$ norms, with the stated non-top hypothesis on the limiting $L^2$ norm.

Original Lean declaration header

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

lemma tendsto_eLpNorm_one_sq_sub
    {μ : MeasureTheory.Measure Ω} {ι : Type*} {l : Filter ι} {a : ι → Ω → ℝ} {b : Ω → ℝ}
    (ha : ∀ n, AEMeasurable (a n) μ) (hb : AEMeasurable b μ)
    (hbfin : MeasureTheory.eLpNorm b 2 μ ≠ ⊤)
    (htend : Filter.Tendsto (fun n => MeasureTheory.eLpNorm (a n - b) 2 μ)
      l (nhds 0)) :
    Filter.Tendsto (fun n => MeasureTheory.eLpNorm (fun ω => (a n ω) ^ 2 - (b ω) ^ 2) 1 μ)
      l (nhds 0)

Module navigation

Module: LevyStochCalc/Brownian/ItoL2CompletionConvergence.

Supporting theorems: eLpNorm_one_mul_le.

Previous: eLpNorm_one_mul_le · Next: condExp_sq_increment_of_martingale.

Source excerpt, lines 134–141 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_ItoL2CompletionConvergence

/-
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/Brownian/ItoL2CompletionConvergence.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
universe u
variable {Ω : Type u} [MeasurableSpace Ω]

Formal statement


lemma LevyStochCalc.Brownian.Ito.tendsto_eLpNorm_one_sq_sub_readable
    {μ : MeasureTheory.Measure Ω} {ι : Type*} {l : Filter ι} {a : ι → Ω → ℝ} {b : Ω → ℝ}
    (ha : ∀ n, AEMeasurable (a n) μ) (hb : AEMeasurable b μ)
    (hbfin : MeasureTheory.eLpNorm b 2 μ ≠ ⊤)
    (htend : Filter.Tendsto (fun n => MeasureTheory.eLpNorm (a n - b) 2 μ)
      l (nhds 0)) :
    Filter.Tendsto (fun n => MeasureTheory.eLpNorm (fun ω => (a n ω) ^ 2 - (b ω) ^ 2) 1 μ)
      l (nhds 0) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/ItoL2CompletionConvergence.lean#L130-L173 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.ItoL2CompletionConvergence`, declaration `LevyStochCalc.Brownian.Ito.tendsto_eLpNorm_one_sq_sub`; 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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