Monotone class theorem: Difference between revisions

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A monotone class in <math>R</math> is a collection <math>\mathcal{M}</math> of [[subsets]] of <math>R</math> which is [[Closure (mathematics)|closed]]{{dn|date=July 2012}} under countable monotone unions and intersections, i.e. if <math>A_i \in \mathcal{M}</math> and <math>A_1 \subset A_2 \subset \ldots </math> then <math>\cup_{i = 1}^\infty A_i \in \mathcal{M}</math>, and similarly for intersections of decreasing sequences of sets.

'''The Monotone Class Theorem''' says that the smallest monotone class containing an [[field of sets|algebra of sets]] <math>\mathcal{G}</math> is precisely the smallest [[Sigma-algebra|σ-algebra]] containing <math>\mathcal{G}</math>.