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

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

References

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