Vanishing Pairings Under L² Convergence

Proved

LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion.integral_mul_eq_zero_of_tendsto_eLpNorm_readable

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

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

This is the continuity of the $L^2$ pairing under convergence in extended $L^2$ norm.

Let $u,v:\Omega\to\mathbb R$ and let $u_n,v_n$ be sequences. Assume $u,v,u_n,v_n$ belong to $L^2(P)$ , that

\int u_nv_n\,dP=0\quad\text{for every }n,

and that $u_n\to u$ and $v_n\to v$ in the sense that

\operatorname{eLpNorm}(u_n-u,2,P)\to0, \qquad \operatorname{eLpNorm}(v_n-v,2,P)\to0.

Then

\int uv\,dP=0.

This auxiliary functional-analytic statement transfers orthogonality from approximating sequences to their $L^2$ limits.

Original Lean declaration header

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

theorem integral_mul_eq_zero_of_tendsto_eLpNorm {u v : Ω → ℝ} {un vn : ℕ → Ω → ℝ}
    (hu : MemLp u 2 P) (hv : MemLp v 2 P)
    (hun : ∀ n, MemLp (un n) 2 P) (hvn : ∀ n, MemLp (vn n) 2 P)
    (h0 : ∀ n, ∫ ω, un n ω * vn n ω ∂P = 0)
    (hcu : Filter.Tendsto (fun n => eLpNorm (fun ω => un n ω - u ω) 2 P) Filter.atTop (nhds 0))
    (hcv : Filter.Tendsto (fun n => eLpNorm (fun ω => vn n ω - v ω) 2 P) Filter.atTop (nhds 0)) :
    ∫ ω, u ω * v ω ∂P = 0

Module navigation

Module: LevyStochCalc/Brownian/CrossOrthogonalityLimit.

Previous: memLp_simpleIntegral · Next: integral_stochasticIntegral_mul_eq_zero.

Source excerpt, lines 332–338 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Brownian_CrossOrthogonalityLimit

/-
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/CrossOrthogonalityLimit.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P] {d : ℕ}
omit [IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion.integral_mul_eq_zero_of_tendsto_eLpNorm_readable {u v : Ω → ℝ} {un vn : ℕ → Ω → ℝ}
    (hu : MemLp u 2 P) (hv : MemLp v 2 P)
    (hun : ∀ n, MemLp (un n) 2 P) (hvn : ∀ n, MemLp (vn n) 2 P)
    (h0 : ∀ n, ∫ ω, un n ω * vn n ω ∂P = 0)
    (hcu : Filter.Tendsto (fun n => eLpNorm (fun ω => un n ω - u ω) 2 P) Filter.atTop (nhds 0))
    (hcv : Filter.Tendsto (fun n => eLpNorm (fun ω => vn n ω - v ω) 2 P) Filter.atTop (nhds 0)) :
    ∫ ω, u ω * v ω ∂P = 0 := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Brownian/CrossOrthogonalityLimit.lean#L329-L368 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Brownian.CrossOrthogonalityLimit`, declaration `LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion.integral_mul_eq_zero_of_tendsto_eLpNorm`; 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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