Monotone class 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 is precisely the smallest š-algebra containing 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 family (i.e. class) of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means has the following properties:
- if and then and
- if and then
Monotone class theorem for sets
Monotone class theorem for sets — Let be an algebra of sets and define to be the smallest monotone class containing Then is precisely the š-algebra generated by ; that is,
Monotone class theorem for functions
Monotone class theorem for functions — Let be a Ļ-system that contains and let be a collection of functions from to with the following properties:
- If then
- If and then and
- 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 which is 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 š-system. By (1) and the Ļāš theorem, Statement (2) implies that contains all simple functions, and then (3) implies that contains all bounded functions measurable with respect to
Results and applications
As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the sigma-ring of
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
- Ļ-š theorem
- Ļ-system – Family of sets closed under intersection
- Dynkin system – Family closed under complements and countable disjoint unions
Citations
- ^ Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 276. ISBN 978-0521765398.
References
- Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020.
Families of sets over | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Is necessarily true of or, is closed under: |
Directed by |
F.I.P. | ||||||||
Ļ-system | ||||||||||
Semiring | Never | |||||||||
Semialgebra (Semifield) | Never | |||||||||
Monotone class | only if | only if | ||||||||
š-system (Dynkin System) | only if |
only if or they are disjoint |
Never | |||||||
Ring (Order theory) | ||||||||||
Ring (Measure theory) | Never | |||||||||
Ī“-Ring | Never | |||||||||
š-Ring | Never | |||||||||
Algebra (Field) | Never | |||||||||
š-Algebra (š-Field) | Never | |||||||||
Dual ideal | ||||||||||
Filter | Never | Never | ||||||||
Prefilter (Filter base) | Never | Never | ||||||||
Filter subbase | Never | Never | ||||||||
Open Topology | (even arbitrary unions) |
Never | ||||||||
Closed Topology | (even arbitrary intersections) |
Never | ||||||||
Is necessarily true of or, is closed under: |
directed downward |
finite intersections |
finite unions |
relative complements |
complements in |
countable intersections |
countable unions |
contains | contains | Finite Intersection Property |
Additionally, a semiring is a Ļ-system where every complement is equal to a finite disjoint union of sets in |