In mathematics, given a positive or a signed measure on a measurable space , a -finite subset is a measurable subset which is the union of a countable number of measurable subsets of finite measure. The measure is called a -finite measure if the set is -finite.
A finite measure, for instance a probability measure, is always -finite.
A different but related notion that should not be confused with -finiteness is s-finiteness.
Let be a measurable space and a measure on it.
The measure is called a ÃÂ-finite measure, if it satisfies one of the four following equivalent criteria:
If is a -finite measure, the measure space is called a -finite measure space.
If is a probability space, then the probability measure, is ÃÂ-finite, because is trivially covered by itself:
For example, Lebesgue measure on the real numbers is not finite, but it is ÃÂ-finite. Indeed, consider the intervals for all integers ; there are countably many such intervals, each has measure 1, and their union is the entire real line.
For Lebesgue measure on a similar disjoint cover can be constructed using unit-volume n-cubes (criterion 2); or by a monotone sequence of expanding n-balls (criterion 3); or by letting in criterion 4 be the multivariate normal density, for which
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 ÃÂ-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. But, the set of natural numbers with the counting measure is ÃÂ -finite.
Locally compact groups which are ÃÂ-compact are ÃÂ-finite under the Haar measure. For example, all connected, locally compact groups G are ÃÂ-compact. To see this, let V be a relatively compact, symmetric (that is V = V<sup>−1</sup>) 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 ÃÂ-finite under Haar measure.
Any non-trivial measure taking only the two values 0 and is clearly non ÃÂ-finite. One example in is: for all , if and only if A is not empty; another one is: for all , if and only if A is uncountable, 0 otherwise. Incidentally, both are translation-invariant.
The class of ÃÂ-finite measures has some very convenient properties; ÃÂ-finiteness can be compared in this respect to separability of topological spaces. Some theorems in analysis require ÃÂ-finiteness as a hypothesis. Usually, both the RadonâÂÂNikodym theorem and Fubini's theorem are stated under an assumption of ÃÂ-finiteness on the measures involved. However, as shown by Irving Segal, they require only a weaker condition, namely localisability.
Though measures which are not ÃÂ-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-ÃÂ-finite if considered as measures on X.
Any ÃÂ-finite measure ü on a space X is equivalent to a probability measure on X: let V<sub>n</sub>, n â N, be a covering of X by pairwise disjoint measurable sets of finite ü-measure, and let w<sub>n</sub>, n â N, be a sequence of positive numbers (weights) such that
The measure ý defined by
is then a probability measure on X with precisely the same null sets as ü.
A Borel measure (in the sense of a locally finite measure on the Borel -algebra) is called a moderate measure iff there are at most countably many open sets with for all and .
Every moderate measure is a -finite measure, the converse is not true.
A measure is called a decomposable measure there are disjoint measurable sets with for all and . For decomposable measures, there is no restriction on the number of measurable sets with finite measure.
Every -finite measure is a decomposable measure, the converse is not true.
A measure is called a s-finite measure if it is the sum of at most countably many finite measures.
Every ÃÂ-finite measure is s-finite, the converse is not true. For a proof and counterexample see relation to ÃÂ-finite measures.