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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:

1. if and then and
2. 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

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

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.