In measure theory and probability, the monotone class theorem connects monotone classes and -algebras. The theorem says that the smallest monotone class containing an algebra of sets is precisely the smallest -algebra containing It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.
Definition of a monotone class
A is a family (i.e. class) of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means has the following properties:
if and then and
if and then
Monotone class theorem for sets
Monotone class theorem for functions
Proof
The following argument originates in Rick Durrett's Probability: Theory and Examples.
Results and applications
As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the -ring of
By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a -algebra.
The monotone class theorem for functions can be a powerful tool that allows statements about particularly simple classes of functions to be generalized to arbitrary bounded and measurable functions.