Monotone class theorem

Let $(\Omega ,{\mathcal {F}})$ be a measure space. A monotone class in $R$ is a collection ${\mathcal {M}}$ of subsets of $R$ which is closed under formation of limits of monotone sequences, 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. Now let ${\mathcal {G}}$ be a field of subsets of $R$ (i.e. ${\mathcal {G}}$ is closed under finite unions and intersections).

The Monotone Class Theorem said that the smallest monotone class containing ${\mathcal {G}}$ coincides with the smallest σ-algebra containing ${\mathcal {G}}$

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.