Monotone class theorem

From Wikipedia, the free encyclopedia

You have new messages (last change).

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.

Retrieved from "http://en.wikipedia.org../../../m/o/n/Monotone_class_theorem.html"
In other languages