Monotone class theorem
In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest monotone class containing an algebra of sets G is precisely the smallest σ-algebra containing G. 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 monotone class in a set is a collection of subsets of which contains and is closed under countable monotone unions and intersections, i.e. if and then , and similarly for intersections of decreasing sequences of sets.
Monotone class theorem for sets
Monotone class theorem for functions
Let be a π-system that contains and let be a collection of functions from to R with the following properties:
(1) If , then
(2) If , then and for any real number
(3) If is a sequence of non-negative functions that increase to a bounded function , then
Then contains all bounded functions that are measurable with respect to , the sigma-algebra generated by
The assumption , (2) and (3) imply that is a λ-system. By (1) and the π − λ theorem, . (2) implies contains all simple functions, and then (3) implies that contains all bounded functions measurable with respect to .
Results and Applications
As a corollary, if G is a ring of sets, then the smallest monotone class containing it coincides with the sigma-ring of G.
By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a sigma-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.