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.

A functional version of this theorem can be found at PlanetMath.[1]

References:

  1. ^ http://planetmath.org/encyclopedia/FunctionalMonotoneClassTheorem.html