Sigma-Finite Measure

A sigma-finite measure is a measure on a measurable space for which has a countable cover by measurable sets of finite measure. Equivalently, there are satisfying

A measure is finite when , which implies sigma-finiteness (Hunter 2011, p. 5).

The converse need not hold. Lebesgue measure on is sigma-finite, since the intervals have finite measure and cover , but its total measure is infinite. Counting measure on a set is sigma-finite iff is countable, since its finite-measure subsets are precisely its finite sets (Hunter 2011, p. 5).