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Monotone class theorem

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

In measure theory and probability, the monotone class theorem connects monotone classes and {{sigma}}-algebras. The theorem says that the smallest monotone class containing an algebra of sets G is precisely the smallest {{sigma}}-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 {{em|{{visible anchor|monotone class}}}} is a family (i.e. class) M of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means M has the following properties:

  1. if A_1, A_2, \ldots \in M and A_1 \subseteq A_2 \subseteq \cdots then {\textstyle\bigcup\limits_{i = 1}^\infty} A_i \in M, and
  2. if B_1, B_2, \ldots \in M and B_1 \supseteq B_2 \supseteq \cdots then {\textstyle\bigcap\limits_{i = 1}^\infty} B_i \in M.

Monotone class theorem for sets

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

Let G be an algebra of sets and define M(G) to be the smallest monotone class containing G. Then M(G) is precisely the {{sigma}}-algebra generated by G; that is \sigma(G) = M(G).

}}

Monotone class theorem for functions

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

Let \mathcal{A} be a {{pi}}-system that contains \Omega\, and let \mathcal{H} be a collection of functions from \Omega to \R with the following properties:

  1. If A \in \mathcal{A} then \mathbf{1}_A \in \mathcal{H} where \mathbf{1}_A denotes the indicator function of A.
  2. If f, g \in \mathcal{H} and c \in \Reals then f + g and c f \in \mathcal{H}.
  3. If f_n \in \mathcal{H} is a sequence of non-negative functions that increase to a bounded function f then f \in \mathcal{H}.

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

}}

=Proof=

The following argument originates in Rick Durrett's Probability: Theory and Examples.{{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}}

{{math proof|drop=hidden|proof=

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

By (1) and the {{pi}}−{{lambda}} theorem, \sigma(\mathcal{A}) \subseteq \mathcal{G}.

Statement (2) implies that \mathcal{H} contains all simple functions, and then (3) implies that \mathcal{H} contains all bounded functions measurable with respect to \sigma(\mathcal{A}).

}}

Results and applications

As a corollary, if G is a 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.

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}}

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References

  • {{Durrett Probability Theory and Examples 5th Edition}}

Category:Families of sets

Category:Theorems in measure theory

fr:Lemme de classe monotone