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.