σ-finite measure

In measure theory, a branch of mathematics, a set in a measure space has σ-finite measure if it is a countable union of sets with finite measure. Similarly, a measure is said to be σ-finite if the set of all points in its corresponding measure space is the countable union of sets with finite measure.

For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. 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, which assigns to each finite set of reals the number of points in the set. This measure is not σ-finite, because every set with finite measure contain only finitely many points, and it would take uncountably many such sets to cover the entire real line.