Parallelogram Distance Criterion for Almost-Everywhere Additivity

This vendored statement gives an $L^2$ parallelogram criterion for identifying a sum almost everywhere.

Let $(\Omega,P)$ be a measurable space with measure $P$, and let $u,v,w:\Omega\to\mathbb R$ belong to $L^2(P)$. Let $A,B,C,D\in[0,\infty]$ be ENNReal values, all strictly below $\infty$. Assume that the squared $L^2$ quantities satisfy

and

Then

for $P$-almost every $\omega$. The statement explicitly uses the finiteness hypotheses to apply toReal to the ENNReal energy identities. This is an auxiliary uniqueness lemma for $L^2$ Itô-integral additivity.

Original Lean declaration header

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

theorem ae_eq_add_of_sq_distances {u v w : Ω → ℝ}
    (hu : MemLp u 2 P) (hv : MemLp v 2 P) (hw : MemLp w 2 P) {A B C D : ℝ≥0∞}
    (hA : A ≠ ⊤) (hB : B ≠ ⊤) (hC : C ≠ ⊤) (hD : D ≠ ⊤)
    (hSu : ∫⁻ ω, (‖u ω‖₊ : ℝ≥0∞) ^ 2 ∂P = A)
    (hSv : ∫⁻ ω, (‖v ω‖₊ : ℝ≥0∞) ^ 2 ∂P = B)
    (hSw : ∫⁻ ω, (‖w ω‖₊ : ℝ≥0∞) ^ 2 ∂P = C)
    (hSwu : ∫⁻ ω, (‖w ω - u ω‖₊ : ℝ≥0∞) ^ 2 ∂P = B)
    (hSwv : ∫⁻ ω, (‖w ω - v ω‖₊ : ℝ≥0∞) ^ 2 ∂P = A)
    (hSuv : ∫⁻ ω, (‖u ω - v ω‖₊ : ℝ≥0∞) ^ 2 ∂P = D)
    (hpar : C + D = 2 * A + 2 * B) :
    w =ᵐ[P] fun ω => u ω + v ω

Module navigation

Module: LevyStochCalc/Brownian/ItoLinear.

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Source excerpt, lines 187–197 · Frozen source 408915f.