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 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 in a set is a collection of subsets of which contains and is closed under countable monotone unions and intersections, i.e. if and then , and similarly for intersections of decreasing sequences of sets.
Monotone class theorem for sets
The following was taken from Probability Essentials, by Jean Jacod and Philip Protter. 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.
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.
Let , i.e. is the smallest monotone class containing . For each set , denote to be the collection of sets such that . It is plain to see that is closed under increasing limits and differences.
Consider . For each , , hence so . This yields when , since is a monotone class containing , and is the smallest monotone class containing
Now, more generally, suppose . For each , we have and by the last result, . Hence, so , and so for all by the argument in the paragraph directly above.
Since for all , it must be that is closed under finite intersections. Furthermore, is closed by differences, so it is also closed under complements. Since is closed under increasing limits as well, it is a sigma-algebra. Since every sigma-algebra is a monotone class, , i.e. is the smallest sigma-algebra containing G
Monotone class theorem for functions
Let be a π-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
The assumption , (2) and (3) imply that is a λ-system. By (1) and the π − 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.
- Jacod, Jean; Protter, Phillip (2004). Probability Essentials. Springer. p. 36. ISBN 978-3-540-438717.
- Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 100. ISBN 978-0521765398.
This article was advanced during a Wikipedia course held at Duke University, which can be found here: Wikipedia and Its Ancestors