Measure

A measure on a sigma-algebra is a nonnegative, countably additive set function that assigns zero to the empty set.

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 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.

A complex measure is a countably additive complex-valued set function on a sigma-algebra. Its total variation norm of a measure is obtained by taking the supremum of sums ${\sum }_j|\nu (E_j)|$ over finite measurable partitions. For finite variation and absolute continuity of measures with respect to a sigma-finite positive measure, the Radon-Nikodym theorem supplies an integrable complex density.

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 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.

The completion of a measure space adjoins every subset of every measurable null set to its sigma-algebra. A statement that two events or σ-algebras agree modulo null sets becomes literal after passing to the corresponding measure-algebra completion.

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

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 $C(K{)}^{\ast }$.

A signed measure is a countably additive real-valued set function, allowing negative values. A finite signed measure has a variation measure, whose total mass is its total variation norm of a measure.

The Dirac probability measure at $x$ satisfies ${\delta }_x(A)=1$ if $x\in A$ and zero otherwise. Its pushforward measure under a measurable map $T$ is $T_{\#}{\delta }_x={\delta }_{T(x)}$.

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.

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.

Counting measure assigns to a set its cardinality, with value infinity for an infinite set.

For $f:\mathbb{N}\to [0,\infty ]$,A complex-valued function is integrable for counting measure exactly when its series is absolutely convergent.

A premeasure is a countably additive nonnegative set function on an algebra of sets, with countable additivity required whenever the union remains in that algebra.

A normalized positive linear functional on $C(\Omega )$ for Cantor space gives a finitely additive set function on the algebra of clopen sets. If a countable disjoint union of such sets is itself clopen, compactness gives a finite subcover; all other terms are empty. Finite additivity therefore gives the required countable additivity, making $\ell$ a premeasure. The Caratheodory extension theorem produces a unique Borel probability measure.

A premeasure on an algebra of sets extends to a measure on the generated sigma-algebra. The extension is unique when the premeasure is sigma-finite.

A positive measure takes values in $[0,\infty ]$, in contrast with signed and complex measures.

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 positive Borel measure on a topological space, the support consists of the points whose every open neighborhood has positive measure. For a finite regular complex measure use its variation measure. Regularity makes the complement of the support null: compact subsets of that open complement have finite covers by null neighborhoods. A regular probability measure on a compact Hausdorff space whose support is a single point is therefore a Dirac measure.

A Borel probability measure is a Borel measure with total mass one. On a compact metric space it is determined by its integrals against continuous functions. On that compact metric space, the family of Borel probability measures is compact for weak convergence of probability measures.

A finite positive Borel measure on a compact Hausdorff space is regular when its values on Borel sets can be approximated from inside by compact sets and from outside by open sets. For signed or complex measures require this property of the variation measure. These are the measures representing the continuous dual space of $C(K)$ in the Riesz-Markov-Kakutani representation theorem.

A Radon measure is a Borel measure that is finite on compact sets and is inner regular, so the measure of a Borel set can be approximated from below by compact subsets. On a locally compact Hausdorff space one also usually includes outer regularity.

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.

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

For a measurable map $f:X\to Y$, the pushforward of $\mu$ is the measureIt is the distribution induced on $Y$ by mapping points distributed according to $\mu$ through $f$.

A homeomorphic embedding of a metric space induces a homeomorphic embedding of its probability measures for their weak topologies. The inverse continuity follows from the open-set Portmanteau theorem, since every relatively open set is the trace of an ambient open set. This does not give surjectivity onto $P(Z)$ when points of $Z$ are outside the embedded image.

If a measure $\nu$ has density $w=d\nu /d\mu$ with respect to $\mu$, thenThis identity is the basis of likelihood ratios and importance sampling.

For a Radon-Nikodym derivative $L_T\ge 0$ with $\mathbb{E}{L}_T=1$, its conditional-expectation process is a nonnegative martingale. If $L_T>0$ almost surely, it defines an equivalent probability measure. A stochastic exponential satisfying the Novikov condition is a common positive density process.

Every locally compact topological group has a nonzero translation-invariant regular measure, unique up to scale. On a compact group it can be normalized to have total mass one.

For pairwise disjoint measurable sets $E_n$, countable additivity means

If $E_n\uparrow E$, then countable additivity gives $\mu (E_n)\uparrow \mu (E)$, even when the limiting measure is infinite.

If $E_n\downarrow E$ and $\mu (E_1)<\infty$, then $\mu (E_n)\downarrow \mu (E)$. Apply continuity from below of a measure to the increasing differences $E_1\setminus E_n$.

A measure $\mu$ on $X$ is finite when $\mu (X)<\infty$.

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.

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