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 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. Definition 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. Examples Lebesgue measure For example, Lebesgue measure on the real numbers is not finite, but it is σ-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. Counting measure 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