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 [[fieldField of sets|algebra of sets]] <math>G</math> is precisely the smallest [[Sigma-algebra|σ{{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 '''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.

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\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\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=

===Statement===

Let <math>G</math> be an [[fieldField 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|σ{{sigma}}-algebra]] generated by <math>G,</math>; i.e.that σis <math>\sigma(G) = M(G).</math>

}}

==Monotone class theorem for functions==

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

===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 '''<math>\R'''</math> with the following properties:

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

(3)# If <math>f_nf, g \in \mathcal{H}</math> isand a<math>c sequence\in of\Reals</math> non-negative functions that increase to a bounded functionthen <math>f + g</math>, thenand <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>

(2) IfThen <math>f,g \in \mathcal{H}</math>, thencontains <math>f+g</math>all bounded functions that are measurable with respect andto <math>cf \in sigma(\mathcal{HA}),</math> forwhich anyis realthe number{{sigma}}-algebra generated by <math>c\mathcal{A}.</math>

}}

===Proof===

(3) 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>

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=100[https://archive.org/details/probabilitytheor00rdur/page/n287 276]|isbn=978-0521765398}}</ref>

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

{{math proof|drop=hidden|proof=

===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.

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

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

<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>

ThenStatement (2) implies that <math>\mathcal{H}</math> contains all boundedsimple functions, thatand arethen measurable(3) withimplies respect tothat <math>\sigma(\mathcal{AH})</math>, thecontains sigma-algebraall generatedbounded byfunctions measurable with respect to <math>\sigma(\mathcal{A}).</math>

}}

==Results and Applicationsapplications==

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 − 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>.

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>

==Results and Applications==

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|{{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|σ-algebra}}

==Citations==

{{reflist|group=note}}

{{reflist}}

==References==

<references/>

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

[[Category:Set families]]

[[Category:SetFamilies familiesof sets]]

[[Category:Theorems in measure theory]]