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

:

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".