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Concept# Σ-finite measure

Summary

In mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞), and a set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measure. A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.e. all finite measures are σ-finite but there are (many) σ-finite measures that are not finite.
A different but related notion that should not be confused with σ-finiteness is s-finiteness.
Let be a measurable space and a measure on it.
The measure is called a σ-finite measure, if it satisfies one of the four following equivalent criteria:
the set can be covered with at most countably many measurable sets with finite measure. This means that there are sets with for all that satisfy .
the set can be covered with at most countably many measurable disjoint sets with finite measure. This means that there are sets with for all and for that satisfy .
the set can be covered with a monotone sequence of measurable sets with finite measure. This means that there are sets with and for all that satisfy .
there exists a strictly positive measurable function whose integral is finite. This means that for all and .
If is a -finite measure, the measure space is called a -finite measure space.
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.

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