“Monotone class theorem”的意思、由来-开放百科全书

词条

Monotone class theorem

释义

Definition of a monotone class
Monotone class theorem for sets
Statement
Monotone class theorem for functions
Statement Proof
Results and applications
References

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 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 class M of sets that is closed under countable monotone unions and intersections, i.e. if and then , and similarly in the other direction.

Monotone class theorem for sets

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 σ-algebra generated by G, i.e. σ(G) = M(G).

Monotone class theorem for functions

Statement

Let be a {{pi}}-system that contains and let be a collection of functions from to R with the following properties:

(1) If , then

(2) If , then and for any real number

(3) If is a sequence of non-negative functions that increase to a bounded function , then

Then contains all bounded functions that are measurable with respect to , the sigma-algebra generated by

Proof

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

^[1]

The assumption , (2) and (3) imply that is a λ-system. By (1) and the {{pi}}−λ theorem, . (2) implies contains all simple functions, and then (3) implies that contains all bounded functions measurable with respect to .

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.

References

1. ^{{cite book|last=Durrett|first=Rick|year=2010|title=Probability: Theory and Examples|edition=4th|publisher=Cambridge University Press|page=100|isbn=978-0521765398}}

Lemme de classe monotone

2 : Set families|Theorems in measure theory

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