Sigma-finite measure

Builds on Measure space Measure Finite measure Countable set

Definition

Let $(X, \Sigma, \mu)$ be a measure space. The measure $\mu$ is σ-finite if there are sets $A_1, A_2, \ldots$ in $\Sigma$ such that $X = \bigcup_{n=1}^{\infty} A_n$ and $\mu(A_n) < \infty$ for every $n$ .

Remarks

Lebesgue measure on $\R$ is σ-finite: take $A_n = [-n, n]$ , of measure $2n$ . Counting measure on the power set of $\R$ is not σ-finite. A set of finite counting measure is finite, a countable union of finite sets is countable, and $\R$ is uncountable. Every finite measure is σ-finite, with $A_n = X$ . The sets $A_n$ can be chosen increasing, or pairwise disjoint, by the usual rearrangements.

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Countable set concept
Finite measure concept
Measure concept
Measure space concept
Sigma-finite measure definition

Tag	`CW5CEF84T2`
URI	`m2://CW5CEF84T2`
Kind	definition
Writing	W0 stub
MSC	28A12
Defines	Sigma-finite measure
Uses	Measure space, Measure, Finite measure, Countable set
Changed	2 November 2026

Sigma-finite measure

Definition

Remarks

Reading path

Further reading

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