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 thatTemplate:Sfnp

• $m(0)=0,m(1)=1$,
• $m(x)>0$ 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

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