Monotone class theorem: Difference between revisions

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Revision as of 08:23, 12 May 2013

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In [[Measure (mathematics)| measure theory]] and [[Probability theory | probability]], the '''monotone class theorem''' connects monotone classes and [[Sigma-algebra|sigma-algebras]]. The theorem says that the smallest monotone class containing an [[field of sets|algebra of sets]] G is precisely the smallest [[Sigma-algebra|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==

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

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===Proof===

The following was taken from Probability Essentials, by Jean Jacod and Philip Protter. <ref name="Jacod">{{cite book|last=Jacod|first=Jean|coauthors=Protter, Phillip |year=2004|title=Probability Essentials|publisher=Springer|page=36|isbn=978-3-540-438717}}</ref> The idea is as follows: we know that the sigma-algebra generated by an algebra of sets G contains the smallest monotone class generated by G. So, we seek to show that the monotone class generated by G is in fact a sigma-algebra, which would then show the two are equal.

To do this, we first construct monotone classes that correspond to elements of G, and show that each equals the M(G), the monotone class generated by G. Using this, we show that the monotone classes corresponding to the other elements of M(G) are also equal to M(G). Finally, we show this result implies M(G) is indeed a sigma-algebra.

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As a corollary, if G is a [[Ring of sets|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.

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==See also==

This article was advanced during a Wikipedia course held at Duke University, which can be found here: [http[Education Program:Duke University//en.wikipedia.org/wiki/Education_ProgramWIKIPEDIA AND ITS ANCESTORS:Duke_University/WIKIPEDIA_AND_ITS_ANCESTORS:_Rethinking_Encyclopedic_Knowledge_in_the_Digital_Age_ Rethinking Encyclopedic Knowledge in the Digital Age (Spring_2013)Spring 2013)|Wikipedia and Its Ancestors]]

[[Category:Set families]]