Monotone class theorem

In measure theory and probability, the monotone class theorem connects monotone classes and 𝜎-algebras.

The theorem says that the smallest monotone class containing an algebra of sets

is precisely the smallest 𝜎-algebra containing

It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.

of sets that is closed under countable monotone unions and also under countable monotone intersections.

has the following properties: Monotone class theorem for sets — Let

be an algebra of sets and define

to be the smallest monotone class containing

is precisely the 𝜎-algebra generated by

Monotone class theorem for functions — Let

contains all bounded functions that are measurable with respect to

The following argument originates in Rick Durrett's Probability: Theory and Examples.

contains all simple functions, and then (3) implies that

contains all bounded functions measurable with respect to

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

By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a 𝜎-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.