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 ${\displaystyle G}$ is precisely the smallest 𝜎-algebra containing ${\displaystyle G.}$ 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) ${\displaystyle M}$ of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means ${\displaystyle M}$ has the following properties:

1. if ${\displaystyle A_{1},A_{2},\ldots \in M}$ and ${\displaystyle A_{1}\subseteq A_{2}\subseteq \cdots }$ then ${\textstyle {\textstyle \bigcup \limits _{i=1}^{\infty }}A_{i}\in M,}$ and
2. if ${\displaystyle B_{1},B_{2},\ldots \in M}$ and ${\displaystyle B_{1}\supseteq B_{2}\supseteq \cdots }$ then ${\textstyle {\textstyle \bigcap \limits _{i=1}^{\infty }}B_{i}\in M.}$

As a corollary, if ${\displaystyle G}$ is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of ${\displaystyle G.}$

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 unionsPages displaying short descriptions of redirect targets
• π-system – Family of sets closed under intersection
• σ-algebra – Algebraic structure of set algebra