Monotone class theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest monotone class containing an algebra of sets G is precisely the smallest σ-algebra containing G. It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.

Definition of a monotone class[edit]

A monotone class in a set R is a collection M of subsets of R which contains R and is closed under countable monotone unions and intersections, i.e. if A_i \in M and A_1 \subset A_2 \subset \ldots then \cup_{i = 1}^\infty A_i \in M, and similarly for intersections of decreasing sequences of sets.

Monotone class theorem for sets[edit]

Statement[edit]

Let G be an algebra of sets and define M(G) to be the smallest monotone class containing G. Then M(G) is precisely the σ-algebra generated by G, i.e. σ(G) = M(G)

Monotone class theorem for functions[edit]

Statement[edit]

Let \mathcal{A} be a π-system that contains \Omega\, and let \mathcal{H} be a collection of functions from \Omega to R with the following properties:

(1) If A \in \mathcal{A}, then \mathbf{1}_{A} \in \mathcal{H}

(2) If f,g \in \mathcal{H}, then f+g and cf \in \mathcal{H} for any real number c

(3) If f_n \in \mathcal{H} is a sequence of non-negative functions that increase to a bounded function f, then f \in \mathcal{H}

Then \mathcal{H} contains all bounded functions that are measurable with respect to \sigma(\mathcal{A}), the sigma-algebra generated by \mathcal{A}

Proof[edit]

The following argument originates in Rick Durrett's Probability: Theory and Examples. [1]

The assumption \Omega\, \in \mathcal{A}, (2) and (3) imply that \mathcal{G} = \{A: \mathbf{1}_{A} \in \mathcal{H}\} is a λ-system. By (1) and the π − λ theorem, \sigma(\mathcal{A}) \subset \mathcal{G}. (2) implies \mathcal{H} contains all simple functions, and then (3) implies that \mathcal{H} contains all bounded functions measurable with respect to \sigma(\mathcal{A}).

Results and Applications[edit]

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

By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a sigma-algebra.

The monotone class theorem for functions can be a powerful tool that allows statements about particularly simple classes of functions to be generalized to arbitrary bounded and measurable functions.

References[edit]

  1. ^ Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 100. ISBN 978-0521765398. 

See also[edit]

This article was advanced during a Wikipedia course held at Duke University, which can be found here: Wikipedia and Its Ancestors