Monotone class theorem

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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 classEdit

A monotone class is a class M of sets that is closed under countable monotone unions and intersections, i.e. if   and   then  , and if   and   then  

Monotone class theorem for setsEdit

StatementEdit

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 functionsEdit

StatementEdit

Let   be a π-system that contains   and let   be a collection of functions from   to R with the following properties:

(1) If  , then  

(2) If  , then   and   for any real number  

(3) If   is a sequence of non-negative functions that increase to a bounded function  , then  

Then   contains all bounded functions that are measurable with respect to  , the sigma-algebra generated by  

ProofEdit

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

The assumption  , (2) and (3) imply that   is a λ-system. By (1) and the πλ theorem,  . (2) implies   contains all simple functions, and then (3) implies that   contains all bounded functions measurable with respect to  .

Results and applicationsEdit

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.

ReferencesEdit

  1. ^ Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 276. ISBN 978-0521765398.