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Measure algebra

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Source updated August 2, 2026

In mathematics, a measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the Boolean algebra of measurable sets modulo null sets.

01Definition

A measure algebra is a Boolean algebra $B$ with a measure $m$ , which is a real-valued function on $B$ such that

$m(0)=0,\ m(1)=1$ ,
$m(x)>0$ if $x\neq 0$ ,
$m(a)\leq m(b)$ for $a\leq b$ ,
If $a_{0},a_{1},a_{2},\dots$ are pairwise disjoint, then

$m{\left(\sum _{n=0}^{\infty }a_{n}\right)}=\sum _{n=0}^{\infty }m(a_{n}).$

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This article is adapted from the Wikipedia article “Measure algebra”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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