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 1Letbe a set. Then, a collectionof subsets ofis a

-system if and only iffor 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 allin.

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.

Lemma 2

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 3For a collectionof 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

(1) |

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)Letbe a-system on a set, andbe a d-system containing. Then,.

In particular, ifis 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 5Letandbe probability measures on a measurable space. If they agree on a-system generating, then.

*Proof:* Let be the set of for which . By equations (1) applied to and , is a d-system. As it also contains a -system generating the -algebra , theorem 4 gives , so on . ⬜

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 6A 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)Letbe an algebra on a set, andbe a monotone class containing. Then,.

In particular, ifis 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)Letbe a nonempty collections of subsets ofsuch thatandare infor all. Then,.

In particular, ifis a monotone class containingthen,.

*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. ⬜

I had some trouble with the last part (starting with “For , the -system property…” of the proof of theorem 4 and it took me a while to figure out the exact mechanics of the arguments. I am proposing a slightly extended explanation below. You can of course completely ignore this but maybe some readers would profit from a slightly more detailed explanation.

For , the -system property immediately gives . Note first that by definition of , we have for any .

As is the minimal d-system generated by and is another d-system, we obtain , which already implies , but so far only for .

Now consider more generally and additionally an . Because of what we just established, we know that , i.e. , which just means that . By symmetry, this implies , thus (because was arbitrary), . By the same argument as above (minimality of the d-system ), this again yields , this time for all .

This statement implies that for all , we know that , i.e. , which is the definition of being a -system. By lemma 3 (make link), is also a -algebra.

[Rest as in the text]

Thanks for your suggestion. I will have a reread through the argument when I have some time, and update accordingly.

The link to Borel is broken.

I meant the first hyperlink to Borel sets.

Fixed. Thanks!

Hello!

First of all, you have an amazing blog, thank you very much for your work!

Apart from that, I have some trouble with the proof of Theorem 8.

Basically, we take a monotone class generated by $\mathcal E$ and prove that it is closed under intersections and complements. However, as far as I understand, one feature is still missing: it is never mentioned that the set $E$ belongs to the corresponding monotone class. In Theorem 7 it holds automatically (due to $\mathcal E$ being an algebra), but how can we say that $E$ is in $M(\mathcal E)$?

Thank you in advance!