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
{\displaystyle B}
-
m
(
0
)
=
0
,
m
(
1
)
=
1
{\displaystyle m(0)=0,\ m(1)=1}
-
m
(
x
)
>
0
{\displaystyle m(x)>0}
if x ≠ 0 {\displaystyle x\neq 0}
-
m
(
a
)
≤
m
(
b
)
{\displaystyle m(a)\leq m(b)}
for a ≤ b {\displaystyle a\leq b}
- If
a
0
,
a
1
,
a
2
,
…
{\displaystyle a_{0},a_{1},a_{2},\dots }
are pairwise disjoint, then
m
(
∑
n
=
0
∞
a
n
)
=
∑
n
=
0
∞
m
(
a
n
)
{\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