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.

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[edit] 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.

[edit] 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.

[edit] 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 = \bigcup_{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.

[edit] Equivalence to a probability measure

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

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

The measure ν defined by

\nu(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.