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For a L2-bounded continuous martingale $M$, define a finite measure on the predictable sigma-algebra by ${\nu }_M(A)=\mathbb{E}{\int }_0^{\infty }{1}_{A}d[M]$. Its total mass is $\mathbb{E}[M{]}_{\infty }=\mathbb{E}{M}_{\infty }^2-\mathbb{E}{M}_0^2$. The Lebesgue space $L^2({\nu }_M)$ identifies predictable processes that agree ${\nu }_M$-almost everywhere. Integration against $M$ is an isometry from this space into L2-bounded continuous martingales starting at zero by the Itô isometry. This gives the correct integrand space even when the quadratic variation is random or is not absolutely continuous in time.

A random measure assigns a measure ${\mu }_{\omega }$ to each outcome so that $\omega \mapsto {\mu }_{\omega }(A)$ is a random variable for every measurable $A$. A finite-variation process gives pathwise signed random measures through Lebesgue–Stieltjes integration.

Every finite positive Borel measure on Euclidean space is inner regular by compact sets and outer regular by open sets. For an open set $U$, the compact sets $\{x:|x|\le n,\mathrm{dist}(x,U^c)\ge 1/n\}$ increase to $U$, so continuity from below of a measure gives inner approximation. The empty complement is handled by compact balls. The class of Borel sets admitting both approximations is closed under complements: interchange inner and outer approximations, then truncate the inner closed set by a large compact ball, using finiteness of the measure. For a countable union, approximate its components from outside with summable errors, and approximate finitely many components from inside after making the remaining union's measure small. Thus this class is a sigma-algebra containing the open sets. In particular, a finite measure carried by a Borel null set has compact subsets of that carrier capturing arbitrarily nearly all its mass.

For uniformly bounded values, an $L^p$ gradient bound with $p>1$, and bounded $(n-1)$-dimensional jump measure, a sequence in the SBV space admits an $L^1$-convergent subsequence whose limit remains special. Gradients converge weakly in $L^p$, and jump measure is lower semicontinuous. These hypotheses prevent diffuse singular derivatives from replacing the controlled interfaces in a minimizing limit.

A measurable set $E$ has finite perimeter in $\Omega$ when ${\chi }_E$ belongs to the BV space. Its relative perimeter is $\mathrm{Per}(E;\Omega )=|D{\chi }_E|(\Omega )$. Relative perimeter excludes the exterior image signal boundary and measures the interior interface in Caccioppoli partitions.

A measure is sigma-finite when its space is a countable union of measurable sets of finite measure.

The spherical derivative compares the mass of shrinking centered open balls with their Lebesgue measure, when the limit exists. This is a density of one measure relative to Euclidean volume, rather than the relative density of a set within one fixed measure. A singular finite Borel measure has spherical derivative zero Lebesgue almost everywhere.

For a finite singular Borel measure, choose compact subsets of a Lebesgue-null carrier whose omitted mass is arbitrarily small. Outside any such compact subset, sufficiently small balls see only the omitted measure. The uncentered maximal weak-type inequality bounds the volume of points where the omitted measure has large density. Letting its total mass tend to zero shows that the upper spherical density is zero almost everywhere. This argument does not require the null carrier itself to be closed or separated from every point outside it.

For a finite positive measure on the real line, this convention for its Stieltjes transform is analytic off the real line and has positive imaginary part in the upper half-plane if the measure is nonzero. Some sources use $(z-x{)}^{-1}$ instead, changing the sign. The transform of an empirical spectral measure equals the normalized trace of the Stieltjes matrix resolvent.

A transport plan is strongly c-monotone if Borel Kantorovich potentials satisfy the displayed feasible inequality everywhere and equality almost everywhere for the plan. Extended values $-\infty$ away from the full marginal-measure sets may be permitted. This is a potential certificate, not a strict version of every cyclic inequality. A finite continuous cost allows the transport potential path construction.

For a smoothed image signal $u_{\sigma }$, average the gradient outer product as $J_{\rho }=G_{\rho }\ast (\nabla u_{\sigma }\nabla u_{\sigma }^T)$. The dominant eigenvector estimates the normal to a coherent image edge, and eigenvalue contrast measures local directional structure. Two distinct scales separate differentiation from orientation averaging.

For a real function in the $L^{\infty }$ unit ball, push uniform measure on $t\in [0,1]$ forward by the displayed sign-valued functions. In the weak-star topology this is a Borel probability concentrated on the extreme points, and Fubini's theorem shows that its barycenter is $f$. Weak-star compactness and separability of the predual $L^1$ place the example within Choquet theorem.

The total variation norm of a finite signed or complex measure is the total mass of its variation measure. Under the Riesz-Markov-Kakutani representation theorem, it equals the norm of the corresponding member of $C(K{)}^{\ast }$.

The same compact set must work simultaneously for every measure in the family. Prokhorov's theorem relates this condition to relative compactness for the weak topology of probability measures on a Polish space.

For a finite signed or complex measure $\mu$, its variation is the positive measureIt controls integration by $|\int fd\mu |\le \int |f|d|\mu |$.