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 $x\ne 0$
• $m(a)\le m(b)$ for $a\le b$
• If $a_0,a_1,a_2,\dots$ are pairwise disjoint, then

$$m\left(\sum _{n=0}^{\mathrm{\infty }}{a}_n\right)=\sum _{n=0}^{\mathrm{\infty }}m(a_n)$$

References

• Jech, Thomas (2003), "Saturated ideals", Set Theory, Springer Monographs in Mathematics (third millennium ed.), Berlin, New York: Springer-Verlag, p. 415, doi:10.1007/3-540-44761-X_22, ISBN 978-3-540-44085-7, https://fa.ewi.tudelft.nl/~hart/onderwijs/set_theory/Jech/22-saturated_ideals.pdf