In mathematics, a positive or a signed measure μ on a set X is called σ-finite if X equals the union of a sequence of measurable sets A1, A2, A3, … of finite measure μ(An) < ∞.
Similarly, a subset of X is called σ-finite if it equals such a countable union.
A different but related notion that should not be confused with σ-finiteness is s-finiteness.
is called a σ-finite measure, if it satisfies one of the four following equivalent criteria: If
[3] For example, Lebesgue measure on the real numbers is not finite, but it is σ-finite.
Indeed, consider the 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.
But, the set of natural numbers
with the counting measure is σ -finite.
Locally compact groups which are σ-compact are σ-finite under the 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.
Any non-trivial measure taking only the two values 0 and
The class of σ-finite measures has 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.
Usually, both the Radon–Nikodym theorem and Fubini's theorem are stated under an assumption of σ-finiteness on the measures involved.
However, as shown by Irving Segal,[4] they require only a weaker condition, namely localisability.
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.
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 The measure ν defined by is then a probability measure on X with precisely the same null sets as μ.
A Borel measure (in the sense of a locally finite measure on the Borel
is called a moderate measure iff there are at most countably many open sets
-finite measure, the converse is not true.
[2] Every σ-finite measure is s-finite, the converse is not true.
For a proof and counterexample see relation to σ-finite measures.