The monotone class theorem, and closely related -system lemma, are simple but fundamental theorems in measure theory, and form an essential step in the proofs of many results. General measurable sets are difficult to describe explicitly so, when proving results in measure theory, it is often necessary to start by considering much simpler sets. The monotone class theorem is then used to extend to arbitrary measurable sets. For example, when proving a result about Borel subsets of , we may start by considering compact intervals and then apply the monotone class theorem. I include this post on the monotone class theorem for reference.
We start with some definitions. Given a collection of subsets of a set , we use the notation for the complement of .
Definition 1 Let be a set. Then, a collection of subsets of is a
- -system if and only if for all .
- algebra if and only if
- for all .
- for all .
- monotone class if and only if
- for all increasing sequences .
- for all decreasing sequences .
- d-system (or Dynkin system, or -system) if and only if
- for all in .
- for all increasing sequences .
- -algebra if and only if
- for all .
- for all sequences .
In short, we can say that a -system is closed under pairwise intersections, an algebra is closed under finite set operations, a monotone class is closed under increasing and decreasing limits, a d-system is closed under increasing limits, taking differences with a subset, and contains the whole space, and a -algebra is closed under countable set operations.
There are various simple relations between the definitions given above.
- every algebra is a -system.
- every d-system is a monotone class.
- every -algebra is a -system, an algebra, a monotone class and a d-system.
Proof: That an algebra is also a -system is clear: for all .
For a set in a d-system , its complement is also in . So, for a decreasing sequence , writing shows that is also a monotone class.
Now, suppose that is a -algebra. For any , define a sequence by and for . Then, is in , so is an algebra and, hence, also a -system. Next, for all . So is a d-system and, hence, also a monotone class. ⬜
The definitions above are also related by the following equivalent characterizations of -algebras.
Lemma 3 For a collection of subsets of , the following are equivalent.
- is a -algebra.
- is both a -system and a d-system.
- is both an algebra and a monotone class.
Proof: Lemma 2 states that a -algebra is also a -system, an algebra, a monotone class, and a d-system. Only the converse statements remain.
Suppose that is both an algebra and a monotone class. If is a sequence in then each finite union is also in . As is increasing, the union is in , which is therefore a -algebra.
Suppose that is both a -system and a d-system. By the d-system property, for all . Hence, for , showing that is an algebra. It is also monotone class, by the second statement of lemma 2. Hence, is a -algebra. ⬜
By definition, measures are constructed with respect to a -algebra on a set . Precisely, a measure satisfies countable additivity
for sequences of pairwise disjoint sets in . In some sense, it would be more natural to define measures on d-systems rather than -algebras. This is because any extension of a probability measure from a collection of sets to the d-system it generates is unique, by the identities
Here, and is an increasing sequence of sets. For -algebras, this does not hold. The measures of and cannot, in general, be determined just from the measures of and . However, only using d-systems would be much too restrictive. Finite unions and intersections are very simple set operations under which we would want the class of measurable sets should be closed. This is where the -system lemma comes in. So long as we start from a -system, then the d-system that it generates is already a -algebra.
Given any collection of subsets of a set , we refer to the smallest -algebra containing as the -algebra generated by , and is denoted by . Similarly, we can talk about the -system, algebra, monotone class, or d-system generated by . The following result is known variously as the -system lemma, Dynkin’s – theorem, or even as the monotone class theorem.
Theorem 4 (Dynkin) Let be a -system on a set , and be a d-system containing . Then, .
In particular, if is the d-system generated by , then .
Proof: First, suppose that is the d-system generated by . As -algebras are also d-systems, the inequality is immediate. Only the reverse inequality needs to be shown.
For any , let denote the collection of sets satisfying . If are in , we have
so . Next, supposing that is an increasing sequence in ,
so . This shows that is itself a d-system
For , the -system property immediately gives , so . Then, for any and , we have or, equivalently, . This, again, shows that . Hence, . Therefore, for all . We have shown that is a -system and, by lemma 3, is a -algebra. So, , giving as required.
Finally, let be any d-system containing . If is the d-system generated by then, as required. ⬜
As an immediate application of the -system lemma, we show uniqueness of extensions of measures.
Lemma 5 Let and be probability measures on a measurable space . If they agree on a -system generating , then .
The cumulative distribution function of a probability measure on the Borel sigma algebra on the real line is defined by . A common use of the above lemma is the following.
Corollary 6 A probability measure on the standard Borel space on the real line is uniquely determined by its distribution function.
Proof: The standard Borel space is , where is the Borel -algebra on . Let consist of the intervals for . This is a -system with . If and are two finite measures with the same distribution function, then for all . So and agree on . Lemma 5 gives . ⬜
I move on to the statement of the monotone class theorem. Here, we weaken the requirements on so that it is only required to be a monotone class, rather than the more restrictive d-system requirement used by the -system lemma. This is at the expense of requiring the stronger condition that is an algebra, and not just a -system.
Theorem 7 (Monotone Class Theorem) Let be an algebra on a set , and be a monotone class containing . Then, .
In particular, if is the monotone class generated by , then .
In many cases where the monotone class theorem is used, the collection of sets that we start with almost, but not quite, satisfies the definition of an algebra. For example, if it is the closed subsets of a metrizable space, then is closed under finite intersections and unions. The complement of a closed set need not be closed, although it is always the limit of an increasing sequence of closed sets. For such cases, the following slightly generalized version can be useful. Note that theorem 7 is an immediate consequence.
We use the notation for the monotone class generated by a collection of subsets of a set .
Theorem 8 (Monotone Class Theorem) Let be a nonempty collections of subsets of such that and are in for all . Then, .
In particular, if is a monotone class containing then, .
Proof: As -algebras are also monotone classes, the inequality is immediate. Only the reverse inequality needs to be shown. Let be the collection of such that is also in . By the condition of the theorem, . If is an increasing sequence then, by the monotone class property, and
are both in . So, is in , showing that it is closed under taking limits of increasing sequences. Similarly, it is closed under taking limits of decreasing sequences so is a monotone class, giving . Therefore, for all .
Now, for any , let denote the collection of such that is also in . For any sequence ,
So, if is an increasing or decreasing sequence, its limit is also in , showing that is a monotone class. By the condition of the theorem, for any , giving . Hence, for all and . This shows that for any . Therefore, , showing that is closed under pairwise intersections.
We have shown that is closed under set complements and pairwise intersections. As it is nonempty, this shows that it is an algebra. By lemma 3, it is a -algebra, giving as required. ⬜