Borel measure

﻿
Borel measure

In mathematics, the Borel algebra is the smallest &sigma;-algebra on the real numbers R containing the
intervals, and the Borel measure is the measure on this &sigma;-algebra which gives to the interval ["a", "b"] the measure "b" − "a" (where "a" < "b").

The Borel measure is not complete, which is why in practice the complete Lebesgue measure is preferred: every Borel measurable set is also Lebesgue measurable, and the measures of the set agree.

In a more general context, let "X" be a locally compact Hausdorff space. A Borel measure is any measure &mu; on the &sigma;-algebra $mathfrak\left\{B\right\}\left(X\right)$ of Borel sets &mdash; the Borel &sigma;-algebra on "X".

If μ is both inner regular and outer regular on all Borel sets, it is called a regular Borel measure.

If μ is inner regular and locally finite, μ is said to be a Radon measure.

References

*
*

Wikimedia Foundation. 2010.

Look at other dictionaries:

• Borel measure — noun A measure whose domain is the Borel σ algebra of a locally compact Hausdorff space …   Wiktionary

• Borel — may refer to: * Émile Borel (1871–1956), a French mathematician * Armand Borel (1923–2003), a Swiss mathematician * Jacques Borel, a French novelist * Gabriel Borel, a French aircraft designer * Borel algebra, operating on Borel sets, named after …   Wikipedia

• Borel set — In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named… …   Wikipedia

• Borel algebra — In mathematics, the Borel algebra (or Borel sigma; algebra) on a topological space X is a sigma; algebra of subsets of X associated with the topology of X . In the mathematics literature, there are at least two nonequivalent definitions of this… …   Wikipedia

• Borel regular measure — In mathematics, an outer measure mu; on n dimensional Euclidean space R n is called Borel regular if the following two conditions hold:* Every Borel set B sube; R n is mu; measurable in the sense of Carathéodory s criterion: for every A sube; R n …   Wikipedia

• Measure (mathematics) — Informally, a measure has the property of being monotone in the sense that if A is a subset of B, the measure of A is less than or equal to the measure of B. Furthermore, the measure of the empty set is required to be 0. In mathematical analysis …   Wikipedia

• measure theory — noun A branch of mathematical analysis, concerned with the theory of integration, that generalizes the intuitive notions of length, area and volume. See Also: Borel measure, complex measure, Haar measur …   Wiktionary

• Borel functional calculus — In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus (that is, an assignment of operators from commutative algebras to functions defined on their spectrum), which has particularly broad… …   Wikipedia

• Borel-Cantelli lemma — In probability theory, the Borel Cantelli lemma is a theorem about sequences of events. In a slightly more general form, it is also a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli.Let ( E n ) be a sequence… …   Wikipedia

• Borel right process — Let E be a locally compact separable metric space.We will denote by mathcal E the Borel subsets of E.Let Omega be the space of right continuous maps from [0,infty) to E that have left limits in E,and for each t in [0,infty), denote by X t the… …   Wikipedia