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 ${\displaystyle B}$ with a measure ${\displaystyle m}$, which is a real-valued function on ${\displaystyle B}$ such that:

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

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

References

• Jech, Thomas (2003), "Saturated ideals" (PDF), 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