Monotone class theorem

A monotone class in R is a collection $\mathcal{M}$ of subsets of R which is closed under countable monotone unions and intersections, i.e. if $A_i \in \mathcal{M}$ and $A_1 \subset A_2 \subset \ldots$ then $\cup_{i = 1}^\infty A_i \in \mathcal{M}$, and similarly for intersections of decreasing sequences of sets.

The Monotone Class Theorem says that the smallest monotone class containing an algebra of sets $\mathcal{G}$ is precisely the smallest σ-algebra containing $\mathcal{G}$.

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

This theorem is used as a type of transfinite induction, and is used to prove many Theorems, such as Fubini's theorem in basic measure theory.