Monotone class theorem explained

G is precisely the smallest -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 is a family (i.e. class)

M of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means M has the following properties:

1. if

A _{1,} A _{2,} \ldots \in M and A _{1} \subseteq A _{2} \subseteq … then $A\_i\; \backslash in\; M,$ and

1. if

B _{1,} B _{2,} \ldots \in M and B _{1} \supseteq B _{2} \supseteq … then $B\_i\; \backslash in\; M.$

Monotone class theorem for functions

Proof

The following argument originates in Rick Durrett's Probability: Theory and Examples.^[1]

Results and applications

As a corollary, if

G is a ring of sets, then the smallest monotone class containing it coincides with the -ring of G.

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.

Notes and References

1. Book: Durrett, Rick. 2010. Probability: Theory and Examples. limited. 4th. Cambridge University Press. 276. 978-0521765398.