Measure algebra

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.

Definition

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

$m (\sum_{n = 0}^{\infty} a_{n}) = \sum_{n = 0}^{\infty} m (a_{n}) .$

References

↑ Jech (2003), p. 415, chapter 22.

Jech, Thomas (2003) [1978]. Set Theory. Springer Monographs in Mathematics (third ed.). Berlin, Heidelberg: Springer. ISBN 978-3-540-44085-7.