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A property holds almost everywhere with respect to a measure when the set of points where it fails has measure zero.

An approximate mean limit of a function is a value satisfying the displayed condition. Functions in the BV space have such finite limits outside an approximate discontinuity set up to negligible sets in the appropriate measure. Their approximate jump points instead have two distinct half-ball limits.

An atom of a measure $\mu$ is a measurable set $A$ of positive measure for which every measurable $B\subseteq A$ has either $\mu (B)=0$ or $\mu (B)=\mu (A)$. A singleton of positive mass is an example. A transport map cannot split the mass of a singleton among several destinations.

The barycenter of a probability measure on a compact convex set is the point satisfying the displayed affine integral identities. It exists by approximating the measure by finitely supported measures and using compactness of their finite convex combinations. It is unique because continuous affine functions separate points. In a Banach-space setting it agrees with the appropriate vector integral when that integral is defined.

A Borel measure on a topological space is a measure defined on the sigma-algebra generated by the open sets, equivalently on the Borel sets.

For a bounded Lipschitz domain, functions can be extended with the displayed bound and common compact support. Reflect across local Lipschitz boundary graphs, combine the reflected functions with a partition of unity, and multiply by a fixed cutoff function. Lipschitz coordinate changes control derivative measures; the product rule bounds cutoff function terms by the $L^1$ norm. Zero extension is also possible with an estimate for BV traces on a hypersurface, but its boundary jump must be included in the variation.

For a nonnegative normalized mollifier supported in the radius-$ϵ$ ball, integrate the BV translation estimate with weight ${\rho }_{ϵ}(h)$. The full-space derivative variation controls the full-space error. Replacing it by variation on a smaller domain is invalid without a support condition: a nonconstant bump outside that domain has zero derivative measure inside but a nonzero mollification error.

The derivative measure of a BV space function decomposes into its absolutely continuous, jump and Cantor parts. The Cantor part of a bounded-variation derivative is singular but not a jump measure on a rectifiable hypersurface. The special bounded-variation space excludes it; finite-length image signal interfaces alone cannot describe it.

Extend the cylinder premeasure from a positive functional to a Borel probability measure and use cylinder-function density in Cantor space. The integral and functional agree on cylinder simple functions, and both are continuous in the supremum norm, so they agree on all of $C(\Omega )$. Uniqueness on the generating cylinder algebra gives uniqueness of the Borel measure.

Maximize the integral of a continuous strictly convex function among probability measures with a fixed barycenter. The supporting measure lemma for affine upper envelopes implies that the maximizing measure has zero integral of the nonnegative envelope gap. Strict convexity makes this gap positive at every nonextreme point, so the measure is concentrated on the extreme boundary. Metrizability makes that boundary Borel and supplies the continuous strictly convex function.

For a scalar function in the BV space, derivative variation is the integral of the relative perimeter measures of its superlevel sets. The formula holds on every Borel subset $A$ of the domain, not just for total mass. It converts the continuous ROF denoising problem into linked minimization problems for finite-perimeter sets.

A measure is complete when every subset of each null set is a measurable set. Such a subset also has measure zero by monotonicity. The completion of a measure extends it to a complete measure by adjoining all subsets of measurable null sets to its sigma-algebra.

For nonnegative integrable $f$, the finite measure $A\mapsto {\int }_{A}fd\mu$ restricted to a sub-sigma-algebra ${\Sigma }_0$ is absolutely continuous with respect to the restricted base measure. Its Radon-Nikodym derivative is ${\Sigma }_0$-measurable and has the required integral identities. Positive and negative parts extend the construction to real integrable functions; real and imaginary parts extend it to complex functions. The result is unique almost everywhere.

The continuum Gaussian free field on a planar domain $D$ is a centered Gaussian process indexed by finite-energy measures or test functions, with covariance given by the Green function of the Laplacian on $D$.

For measurable sets $E_n$, a measure satisfiesReplace $E_n$ by the disjoint sets $E_n\setminus {\bigcup }_{j<n}{E}_j$, use countable additivity, and then monotonicity. The same property is an axiom for an outer measure on arbitrary sets.

A density ratio compares two probability density functions relative to the same underlying measure. More generally it is a Radon-Nikodym derivative when one probability measure has absolute continuity of measures with respect to the other.