Monotone class theorem

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A monotone class in R is a collection of subsets of R which is closed under countable monotone unions and intersections, i.e. if and then , and similarly for intersections of decreasing sequences of sets.

The Monotone Class Theorem says that the smallest monotone class containing an algebra of sets coincides with the smallest σ-algebra containing .

As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the sigma-ring of .

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.