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}}-algebrasalgebra]]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:

==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 G. Then M(G) is precisely the [[Sigma-algebra|σ-algebra]] generated by G, i.e. σ(G) = M(G)

# 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

===Proof===

# 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>

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.

==Monotone class theorem for sets==

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.

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

Let <math>\mathcal{B} = M(G)</math>, i.e. <math>\mathcal{B}</math> is the smallest monotone class containing <math>G</math>. For each set <math>B</math>, denote <math>\mathcal{B}_B</math> to be the collection of sets <math>A \in \mathcal{B}</math> such that <math>A \cap B \in \mathcal{B}</math>. It is plain to see that <math>\mathcal{B}_B</math> is closed under increasing limits and differences.

{{math theorem|name=Monotone class theorem for functions|note=|style=|math_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 real-valued functions from <math>\Omega</math> to <math>\R</math> with the following properties:

Consider# <math>B \in G</math>. For eachIf <math>CA \in G</math>, <math> B \cap C \in G \subset \mathcal{BA}</math>, hencethen <math>C \in \mathcalmathbf{B1}_B </math> so <math>G \subset \mathcal{B}_B </math>. This yields <math>\mathcal{B}_B = \mathcal{B}</math> when <math> B_A \in G</math>, since <math>\mathcal{BH}_B</math> is a monotone class containing <math>G</math>,where <math>\mathcal{B}_B \subset \mathcalmathbf{B1}_A</math> and <math>\mathcal{B}</math> isdenotes the smallest[[indicator monotonefunction]] class containingof <math>GA.</math>

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

(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>

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===

Now, more generally, suppose <math>B \in \mathcal{B}</math>. For each <math>C \in G</math>, we have <math> B \in \mathcal{B}_C</math> and by the last result, <math>B \cap C \in \mathcal{B}</math>. Hence, <math>C \in \mathcal{B}_B</math> so <math>G \subset \mathcal{B}_B</math>, and so <math>\mathcal{B}_B = \mathcal{B}</math> for all <math>B \in \mathcal{B}</math> by the argument in the paragraph directly above.

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>

{{math proof|drop=hidden|proof=

Since <math>\mathcal{B}_B = \mathcal{B}</math> for all <math>B \in \mathcal{B}</math>, it must be that <math>\mathcal{B}</math> is closed under intersections. Furthermore, <math>\mathcal{B}</math> is closed by differences, so it is also closed under complements. Since <math>\mathcal{B}</math> is closed under increasing limits as well, it is a sigma-algebra. Since every sigma-algebra is a monotone class, <math>\mathcal{B} = \sigma\,(G)</math>, i.e. <math>\mathcal{B}</math> is the smallest sigma-algebra containing G

(1)The Ifassumption <math>A\Omega\, \in \mathcal{A},</math> (2), thenand (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}) \subseteq \mathcal{G}.</math>

The assumption <math>\Omega\, \in \mathcal{A}</math>,Statement (2) and (3) implyimplies 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>

}}

==Results and Applicationsapplications==

==Monotone class theorem for functions==

===Statement===

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

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>

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

(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>

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.

(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>

==See also==

* {{annotated link|Dynkin system}}

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>

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

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

* {{annotated link|σ-algebra}}

===Proof=Citations==

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|edition=4th|publisher=Cambridge University Press|page=100|isbn=978-0521765398}}</ref>

{{reflist|group=note}}

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>

{{reflist}}

==References==

==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.

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

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.

==References==

<references/>

[[Category:SetFamilies familiesof sets]]

[[Category:Theorems in measure theory]]