SIGMA-FINITE MEASURE

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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.

Contents

Examples

Properties

Locally compact groups

Equivalence to a probability 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} mu (A cap V_{n})
is then a probability measure on ''X'' with precisely the same null sets as ''μ''.