Σ-finite measure

Σ-finite measure

In mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set "X" is called finite, if μ("X") is a finite real number (rather than ∞). The measure μ is called σ-finite, if "X" is the countable union of measurable sets of finite measure. A set in a measure space has σ-finite measure, if it is a union of sets with finite measure.

Examples

For example, Lebesgue measure on the real numbers is not finite, but it is "σ"-finite. Indeed, consider the closed intervals ["k","k+1"] for all integers "k"; there are countably many such intervals, each has measure 1, and their union is the entire real line.

Alternatively, consider the real numbers with the counting measure; the measure of any finite set is the number of elements in the set, and the measure of any infinite set is infinity. This measure is not "σ"-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line.

Properties

The class of "σ"-finite measures have some very convenient properties; "σ"-finiteness can be compared in this respect to separability of topological spaces. Some theorems in analysis require "σ"-finiteness as a hypothesis. For example, both the Radon-Nikodym theorem and the Fubini theorem are invalid without an assumption of "σ"-finiteness (or something similar) on the measures involved.

Though measures which are not "σ"-finite are sometimes regarded as pathological, they do in fact occur quite naturally. For instance, if "X" is a metric space of Hausdorff dimension "r", then all lower dimensional Hausdorff measures are non-"σ"-finite if considered as measures on "X".

Locally compact groups

Locally compact groups which are "σ"-compact are "σ"-finite under Haar measure. For example, all connected, locally compact groups "G" are "σ"-compact. To see this, let "V" be a relatively compact, symmetric (that is "V" = "V"-1) open neighborhood of the identity. Then

: H = igcup_{n in mathbb{N V^n

is an open subgroup of "G". Therefore "H" is also closed since its complement is a union of open sets and by connectivity of "G", must be "G" itself. Thus all connected Lie groups are "σ"-finite under Haar measure.

Equivalence to a probability measure

Any "σ"-finite measure "μ" on a space "X" is equivalent to a probability measure on "X": let "V""n", "n" ∈ N, be a covering of "X" by pairwise disjoint measurable sets of finite "μ"-measure, and let "w""n", "n" ∈ N, be a sequence of positive numbers (weights) such that

:sum_{n = 1}^{infty} w_{n} = 1.

The measure "ν" defined by

: u(A) = sum_{n = 1}^{infty}w_n mu (A cap V_{n})

is then a probability measure on "X" with precisely the same null sets as "μ".


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Locally finite measure — In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.DefinitionLet ( X , T ) be a Hausdorff topological space and let Sigma; be a sigma; algebra on X that contains… …   Wikipedia

  • σ-finite measure — noun That it is expressible as a countable union of sets of finite measure …   Wiktionary

  • 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

  • Finite-dimensional distribution — In mathematics, finite dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the projection of a measure (or process) onto a finite dimensional vector space (or… …   Wikipedia

  • Finite strain theory — Continuum mechanics …   Wikipedia

  • measure — measurer, n. /mezh euhr/, n., v., measured, measuring. n. 1. a unit or standard of measurement: weights and measures. 2. a system of measurement: liquid measure. 3. an instrument, as a graduated rod or a container of standard capacity, for… …   Universalium

  • Measure-preserving dynamical system — In mathematics, a measure preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Contents 1 Definition 2 Examples 3 Homomorphisms 4 …   Wikipedia

  • Measure word — This article is about measure words in general. For measure words in Chinese, see Chinese classifier. In linguistics, measure words are words (or morphemes) that are used in combination with a numeral to indicate an amount of some noun. They… …   Wikipedia

  • Radon measure — In mathematics (specifically, measure theory), a Radon measure, named after Johann Radon, is a measure on the σ algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular. Contents 1 Motivation 2 Definitions …   Wikipedia

  • There is no infinite-dimensional Lebesgue measure — In mathematics, it is a theorem that there is no analogue of Lebesgue measure on an infinite dimensional space. This fact forces mathematicians studying measure theory on infinite dimensional spaces to use other kinds of measures: often, the… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”