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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.

A ** monotone class** 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** — Let be an algebra of sets and define to be the smallest monotone class containing Then is precisely the 𝜎-algebra generated by ; that is

**Monotone class theorem for functions** — Let be a π-system that contains and let be a collection of functions from to with the following properties:

- If then where denotes the indicator function of
- If and then and
- 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 which is the 𝜎-algebra generated by

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

The assumption (2), and (3) imply that is a 𝜆-system. By (1) and the π−𝜆 theorem, Statement (2) implies that contains all simple functions, and then (3) implies that contains all bounded functions measurable with respect to

As a corollary, if 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.

- Dynkin system – Family closed under complements and countable disjoint unions
- π-𝜆 theorem – Family closed under complements and countable disjoint unions
- π-system – Family of sets closed under intersection
- σ-algebra – Algebraic structure of set algebra

- Durrett, Richard (2019).
*Probability: Theory and Examples*(PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020.