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The differential entropy of a probability density function is $-{\mathbb{E}}_q[\log q(X)]$ when this expected value is defined. Unlike discrete information entropy, it can be negative and depends on the coordinates and reference measure. It is the entropy term in the evidence lower bound for continuous variational inference.

If $E\subseteq [0,1]$ is a Lebesgue measurable set and $0\le b\le \lambda (E)$, some measurable subset $B\subseteq E$ has Lebesgue measure $b$. The function $F(t)=\lambda (E\cap [0,t])$ has Lipschitz continuity with constant $1$, starts at $0$ and ends at $\lambda (E)$; the intermediate value theorem proves the assertion. Applying the assertion successively to remaining portions of $E$ gives pairwise disjoint measurable pieces with any finite list of nonnegative measures whose sum is at most $\lambda (E)$. This is a concrete consequence of non-atomic measure structure, which fails for general measures containing an atom of a measure.

A minimizer of the Mumford–Shah functional with bounded image signal data has an essentially closed jump set: replacing $J_u$ by its relative closure adds no $(n-1)$-dimensional measure. Its complement supports a Sobolev representative. This regularity theorem connects a relaxed SBV space minimizer to the classical closed-edge formulation; it is not a property of every special bounded-variation function.

For real scalars the extremes are the almost-everywhere sign-valued functions. A positive-measure region uniformly inside $[-1,1]$ permits opposite nontrivial bounded perturbations, proving necessity. Equality at either endpoint forces both members of a convex decomposition to agree, proving sufficiency. For complex scalars the same criterion uses unit-modulus functions and the strict convexity of the complex unit disk.

An optimal transport plan of finite cost for a finite continuous cost function is c-cyclically monotone. A strict violation at finitely many support points persists in small product neighborhoods; subtracting small normalized pieces and inserting cyclically re-paired product measures preserves both marginals and strictly improves the finite cost. The finite-value hypothesis matters: if every competitor has infinite cost, a nonmonotone plan may still be an extended-value minimizer.

A finite-measure set is a measurable set on which the ambient measure is finite. Restricting to such a set makes its indicator function belong to every Lp space with finite exponent, enabling local Radon-Nikodym theorem arguments even when the whole measure space is not sigma-finite.

Let $h:\mathbb{R}\to \mathbb{R}$ be continuous, strictly increasing, and satisfy Lusin condition N. Thendefines a measure on the Lebesgue measurable sets, and $\nu \ll \lambda$. Moreover,for $a\le b$.

A real or complex measurable function is integrable for a measure $\mu$ when its absolute value has finite integral. For a random variable, this is integrability with respect to its probability measure, and makes its expectation finite.

For a measurable transformation $T$, a measure is invariant if the pushforward measure equals the original measure, or equivalently $\mu (T^{-1}A)=\mu (A)$ for every measurable $A$. This is the measure used in a measure-preserving system.

A subset of $X$ is measurable relative to a sigma-algebra $\mathcal{F}$ when it belongs to $\mathcal{F}$. A measure defined on $\mathcal{F}$ assigns sizes to these sets. Measurability depends on the chosen sigma-algebra; for example, a Lebesgue measurable set need not be a Borel set.

A measurable space is a set equipped with a sigma-algebra of measurable subsets. A measure space additionally specifies a measure on that sigma-algebra.

When source and target have continuous strictly increasing cumulative distributions, all transport maps are obtained by choosing a Lebesgue-measure-preserving map $S$ on the unit interval. The maps $F$ and $G$ turn the two measures into uniform measure. The choice $S=\mathrm{id}$ gives the monotone rearrangement.

A measure space is a set $X$, a sigma-algebra $\mathcal{M}$ on it, and a measure $\mu$ defined on $\mathcal{M}$.

A non-atomic measure contains no atom of a measure. For a finite Borel measure on a Polish space, this is equivalent to giving every singleton mass zero. For a probability measure on $\mathbb{R}$, it is also equivalent to continuity of its cumulative distribution function.

For bounded BV space data and $s<t$, nested ROF level-set formulation minimizing sets cannot share an interface of positive surface measure away from data jumps. On regular graph patches, upward and downward first variations use the BV traces on a hypersurface. At contacts outside $J_f$ the forcing BV trace on a hypersurface agrees, whereas the two levels differ by $t-s$. Graph regularity and contact differentiation give the contradiction. Bounded forcing is a hypothesis of this argument; arbitrary unbounded $L^2\cap BV$ data are not covered by this statement.

A measurable set $N$ is a null set for a measure $\mu$ when $\mu (N)=0$. A property holds almost everywhere when its exceptions lie in a null set. With a complete measure, every subset of a null set is measurable and is itself a null set.

A measure on a topological space is outer regular on its measurable sets when the displayed open-cover identity holds. Lebesgue measure has this property. It allows a null set to be placed in arbitrarily small open neighborhoods, and a compact small set to receive a smooth cutoff of similarly small integral.