Energy Triangle Inequality for Three Integrands

Proved

LevyStochCalc.Brownian.Ito.lintegral_energy_sub_le_readable

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

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

This vendored inequality controls the squared energy of a difference through a common third integrand.

Let $f,g,h:\Omega\times\mathbb R\to\mathbb R$ be jointly measurable, let $P$ be a measure on $\Omega$ , and let $T\in\mathbb R$ . Then

\int_\Omega\int_{[0,T]} \|f(\omega,s)-g(\omega,s)\|^2\,ds\,dP(\omega) \leq 2\left( \int_\Omega\int_{[0,T]}\|f-h\|^2\,ds\,dP + \int_\Omega\int_{[0,T]}\|g-h\|^2\,ds\,dP \right).

The integrals are expressed as ENNReal lower integrals, so the estimate also covers infinite values.

This is a reusable quadratic comparison estimate for coefficient energies.

Original Lean declaration header

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

theorem lintegral_energy_sub_le {f g h : Ω → ℝ → ℝ}
    (hf : Measurable (Function.uncurry f)) (hg : Measurable (Function.uncurry g))
    (hh : Measurable (Function.uncurry h)) (T : ℝ) :
    ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖f ω s - g ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P
      ≤ 2 * ((∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
            (‖f ω s - h ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P)
          + ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
            (‖g ω s - h ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P)

Module navigation

Module: LevyStochCalc/Ito/VectorItoVersionSup.

Previous: lintegral_iSup_sq_norm_version_sub_le.

Source excerpt, lines 378–385 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_LevyStochCalc_Ito_VectorItoVersionSup

/-
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/VectorItoVersionSup.lean; Apache-2.0; provenance in formalization/vendor/vendor-lock.json of the KLS repository)
open MeasureTheory ProbabilityTheory LevyStochCalc.Probability
open scoped NNReal ENNReal
universe u
variable {Ω : Type u} [MeasurableSpace Ω] {P : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure P]
  {n d : ℕ} (W : LevyStochCalc.Brownian.Multidim.MultidimBrownianMotion P d)
  (ℱ : MeasureTheory.Filtration ℝ ‹MeasurableSpace Ω›)
  (hcoord : ∀ k : Fin d, LevyStochCalc.Brownian.IsBrownianFiltration (W.W k) ℱ)
omit [IsProbabilityMeasure P]

Formal statement


theorem LevyStochCalc.Brownian.Ito.lintegral_energy_sub_le_readable {f g h : Ω → ℝ → ℝ}
    (hf : Measurable (Function.uncurry f)) (hg : Measurable (Function.uncurry g))
    (hh : Measurable (Function.uncurry h)) (T : ℝ) :
    ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T, (‖f ω s - g ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P
      ≤ 2 * ((∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
            (‖f ω s - h ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P)
          + ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
            (‖g ω s - h ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P) := by sorry

Source

Lean source: https://github.com/cgarryZA/LevyStochCalc/blob/984dbf0b790031eac05d56fe5a84b65e15e1e932/LevyStochCalc/Ito/VectorItoVersionSup.lean#L375-L418 (repository human-ai-mathematics/kls-lean, commit 408915f, module `LevyStochCalc.Ito.VectorItoVersionSup`, declaration `LevyStochCalc.Brownian.Ito.lintegral_energy_sub_le`; 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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