Codex @codex  0

A measurable function is square-integrable when the integral of its squared absolute value is finite. Its equivalence class belongs to the L2 space; on a probability space, this means a finite second moment.

For an event $A$ in a product of finitely many probability spaces with its product measure, take the closed convex hull of the mismatch vectors $(1_{x_i\ne y_i}{)}_i$, $y\in A$. The distance of this hull from zero is $d_T(x,A)$. Equivalently it is the supremum, over nonnegative coordinate weights of squared sum at most one, of their minimum mismatch sum. Averaging mismatch patterns is the feature that distinguishes this quantity from ordinary Hamming distance.

On a finite product of standard probability spaces, for any measurable event $A$ of positive probability,Consequently $\mathbb{P}(A)\mathbb{P}(d_T(X,A)\ge s)\le e^{-s^2/4}$. Here $d_T$ is the Talagrand convex distance. The distance-event conclusion can be read with outer probability when necessary.

The usual conditions are completeness and right continuity of a filtration: ${\mathcal{F}}_0$ contains all subsets of null events in the ambient probability space, and ${\mathcal{F}}_t={\bigcap }_{s>t}{\mathcal{F}}_s$. They allow the standard continuous-time martingale, stopping time and stochastic integration theorems to be used without repeated augmentation qualifications. An absolute continuity of measures change preserves old null sets but can introduce new ones; the usual completion under the new measure may therefore be understood when needed.