In mathematics, a measure algebra is a
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...
with a countably additive positive measure. A
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...
on a
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
gives a measure algebra on the Boolean algebra of measurable sets modulo
null set
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length.
The notion of null s ...
s.
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''≠0
*''m'' is countably additive: ''m''(Σ''x''
''i'') = Σ''m''(''x''
''i'') if the ''x''
''i'' are a countable set of elements that are disjoint (''x''
''i'' ∧ ''x''
''j''=0 whenever ''i''≠''j'').
References
*{{Citation , last1=Jech , first1=Thomas , author1-link=Thomas Jech , title=Set Theory , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , edition=third millennium , series=Springer Monographs in Mathematics , isbn=978-3-540-44085-7 , doi=10.1007/3-540-44761-X_22 , year=2003, chapter=Saturated ideals, page=415
Measure theory