Monotone class theorem

In measure theory and probability, the monotone class theorem connects monotone classes and $\sigma$-algebras. The theorem says that the smallest monotone class containing an algebra of sets $$G$$ is precisely the smallest 𝜎-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 $$A_1 \subseteq A_2 \subseteq \cdots$$ then ${\textstyle\bigcup\limits_{i = 1}^\infty} A_i \in M,$  and
 * 2) if $$B_1, B_2, \ldots \in M$$ and $$B_1 \supseteq B_2 \supseteq \cdots$$ then ${\textstyle\bigcap\limits_{i = 1}^\infty} B_i \in M.$

Monotone class theorem for sets
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Monotone class theorem for functions
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Proof
The following argument originates in Rick Durrett's Probability: Theory and Examples.

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Results and applications
As a corollary, if $$G$$ is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of $$G.$$

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.