Monotone class theorem: Difference between revisions

Revision as of 10:54, 12 November 2017

Revision as of 01:44, 14 February 2018

Line 1:

In [[Measure (mathematics)|measure theory]] and [[Probability theory|probability]], the '''monotone class theorem''' connects monotone classes and [[sigma-algebra]]s. 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''' is a collection <math>''M</math>'' of set-theoretic [[Class (set theory)|classes]] which 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 \ldotscdots</math> then <math>\cup_bigcup_{i = 1}^\infty A_i \in M</math>, and similarly in the other direction.

==Monotone class theorem for sets==

===Statement===

Let ''G'' be an [[field of sets|algebra of sets]] and define ''M''(''G'') to be the smallest monotone class containing &nbsp;''G''. Then ''M''(''G'') is precisely the [[Sigma-algebra|''σ''-algebra]] generated by ''G'', i.e. &nbsp;''σ''(''G'') &nbsp;= &nbsp;''M''(''G'').

==Monotone class theorem for functions==

===Statement===

Let <math>\mathcal{A}</math> be a [[Pi system|π{{pi}}-system]] that contains <math>\Omega\,</math> and let <math>\mathcal{H}</math> be a collection of functions from <math>\Omega</math> to '''R''' with the following properties:

(1) If <math>A \in \mathcal{A}</math>, then <math>\mathbf{1}_{A}_A \in \mathcal{H}</math>

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

Line 26:

<ref name="Durrett">{{cite book|last=Durrett|first=Rick|year=2010|title=Probability: Theory and Examples|edition=4th|publisher=Cambridge University Press|page=100|isbn=978-0521765398}}</ref>

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

==Results and Applicationsapplications==

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 &nbsp;''G''.

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