Monotone class theorem: Difference between revisions

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{{Short description|Measure theory and probability theorem}}

In [[Measure (mathematics)|measure theory]] and [[Probability theory|probability]], the '''monotone class theorem''' connects monotone classes and [[Sigma-algebra|{{sigma}}-algebra]]s. The theorem says that the smallest [[#Definition of a monotone class|monotone class]] containing an [[Field of sets|algebra of sets]] <math>G</math> is precisely the smallest [[Sigma-algebra|{{sigma}}-algebra]] containing&nbsp;<math>G.</math> 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 '''''{{em|{{visible anchor|monotone class''}}}}''' is a [[Family of sets|family]] (i.e. class) <math>M</math> of sets that is [[Closure (mathematics)|closed]] under countable monotone unions and also under countable monotone intersections. Explicitly, this means <math>M</math> has the following properties:

# if <math>A_1, A_2, \ldots \in M</math> and <math>A_1 \subseteq A_2 \subseteq \cdots</math> then <math display="inline">{\textstyle\bigcup\bigcup_limits_{i = 1}^{\infty} A_i \in M,</math> and

# if <math>B_1, B_2, \ldots \in M</math> and <math>B_1 \supseteq B_2 \supseteq \cdots</math> then <math display="inline">{\textstyle\bigcap\bigcap_limits_{i = 1}^{\infty} B_i \in M.</math>

== Monotone class theorem for sets ==

{{math theorem|name=Monotone class theorem for sets|note=|style=|math_statement=

Let <math>G</math> be an [[Field of sets|algebra of sets]] and define <math>M(G)</math> to be the smallest monotone class containing <math>G.</math> Then <math>M(G)</math> is precisely the [[Sigma-algebra|{{sigma}}-algebra]] generated by <math>G</math>; that is, <math>\sigma(G) = M(G).</math>

}}

== Monotone class theorem for functions ==

{{math theorem|name=Monotone class theorem for functions|note=|style=|math_statement=

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

# If <math>A \in \mathcal{A}</math> then <math>\mathbf{1}_A \in \mathcal{H}</math> where <math>\mathbf{1}_A</math> denotes the [[indicator function]] of <math>A.</math>

# If <math>f, g \in \mathcal{H}</math> and <math>c \in \RReals</math> then <math>f + g</math> and <math>c f \in \mathcal{H}.</math>

# If <math>f_n \in \mathcal{H}</math> is a sequence of non-negative functions that increase to a bounded function <math>f</math> then <math>f \in \mathcal{H}.</math>

Then <math>\mathcal{H}</math> contains all bounded functions that are measurable with respect to <math>\sigma(\mathcal{A}),</math> which is the {{sigma}}-algebra generated by <math>\mathcal{A}.</math>

}}

=== Proof ===

The following argument originates in [[Rick Durrett]]'s Probability: Theory and Examples.<ref name="Durrett">{{cite book|last=Durrett|first=Rick|year=2010|title=Probability: Theory and Examples|url=https://archive.org/details/probabilitytheor00rdur|url-access=limited|edition=4th|publisher=Cambridge University Press|page=[https://archive.org/details/probabilitytheor00rdur/page/n287 276]|isbn=978-0521765398}}</ref>

{{math proof|drop=hidden|proof=

The assumption <math>\Omega\, \in \mathcal{A},</math> (2), and (3) imply that <math>\mathcal{G} = \left\{ A : \mathbf{1}_{A} \in \mathcal{H} \right\}</math> is a {{lambda}}-system.

By (1) and the [[Dynkin system|{{pi}}−{{lambda}} theorem]], <math>\sigma(\mathcal{A}) \subsetsubseteq \mathcal{G}.</math>

Statement (2) implies that <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 applications ==

As a corollary, if <math>G</math> is a [[Ringring of sets|ring]] of sets, then the smallest monotone class containing it coincides with the [[Sigma-ring|{{sigma}}-ring]] of <math>G.</math>

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

== See also ==

* {{annotated link|Dynkin system}}

* {{annotated link|π-λ theorem|{{pi}}-{{lambda}} theorem}}

* {{annotated link|Pi-system|{{pi}}-system}}

* {{annotated link|Dynkin systemσ-algebra}}

== Citations ==

{{reflist|group=note}}

{{reflist}}

== References ==

* {{Durrett Probability Theory and Examples 5th Edition}} <!-- {{sfn|Durrett|2019|p=}} -->

[[Category:SetFamilies familiesof sets]]

[[Category:Theorems in measure theory]]