Consider a probability space and a subset *S* of . The projection is the set of such that there exists a with . We can ask whether there exists a map

such that . From the definition of the projection, values of satisfying this exist for each individual . By invoking the axiom of choice, then, we see that functions with the required property do exist. However, to be of use for probability theory, it is important that should be measurable. Whether or not there are measurable functions with the required properties is a much more difficult problem, and is answered affirmatively by the *measurable selection* theorem. For the question to have any hope of having a positive answer, we require *S* to be measurable, so that it lies in the product sigma-algebra , with denoting the Borel sigma-algebra on . Also, less obviously, the underlying probability space should be complete. Throughout this post, will be assumed to be a complete probability space.

It is convenient to extend to the whole of by setting for outside of . Then, is a map to the extended nonnegative reals for which precisely when is in . Next, the *graph* of , denoted by , is defined to be the set of with . The property that whenever is expressed succinctly by the inclusion . With this notation, the measurable selection theorem is as follows.

Theorem 1 (Measurable Selection)For any , there exists a measurable such that and

(1)

As noted above, if it wasn’t for the measurability requirement then this theorem would just be a simple application of the axiom of choice. Requiring to be measurable, on the other hand, makes the theorem much more difficult to prove. For instance, it would not hold if the underlying probability space was not required to be complete. Note also that, stated as above, measurable selection implies that the projection of *S* is equal to a measurable set , so the measurable projection theorem is an immediate corollary. I will leave the proof of Theorem 1 for a later post, together with the proofs of the section theorems stated below.

A closely related problem is the following. Given a measurable space and a measurable function, , does there exist a measurable right-inverse on the image of ? This is asking for a measurable function, , from to such that . In the case where is the Borel space , Theorem 1 says that it does exist. If *S* is the graph then will be the required right-inverse. In fact, as all uncountable Polish spaces are Borel-isomorphic to each other and, hence, to , this result applies whenever is a Polish space together with its Borel sigma-algebra.

Continuous-time stochastic calculus is generally applied under the setting of a filtered probability space which, as usual, I am assuming to be complete. It can be important to choose the random time given by Theorem 1 to be a stopping time. For this to be possible, the stronger measurability criterion that *S* is in the optional sigma-algebra is required.

Theorem 2 (Optional Section)For any and , there exists a stopping time such that and

(2)

In the statement of the optional section theorem, the equality (1) has been replaced by an inequality (2). The condition ensures that is a subset of , and the measure of the difference of these events is

Inequality (2) states that this can be made as small as we like and, so, is an approximate form of equality (1).

In fact, it is not generally possible to choose the stopping time to satisfy (1). To see this, consider the following example. Let be a random variable taking values in and such that for each positive *t*. For example, take to be uniformly distributed on . Let be the (completed) natural filtration of , so that is a stopping time, and set . It can be seen that is generated by the sets over and, hence, is trivial when restricted to . So, every -measurable random variable is almost-surely constant on the event . Now, any stopping time can be seen to be deterministic on the event . So, if , we have

almost surely, for some fixed positive *s*. Then,

Next, for a predictable set, , it is possible to do better than the optional section theorem. The stopping time can be chosen to be predictable.

Theorem 3 (Predictable Section)For any and , there exists a predictable stopping time such that and

Again, it is not generally possible to choose to satisfy (1). With the stopping time and filtration as in the example above, consider taking . Let be a predictable stopping time with . If is a sequence of stopping times announcing then, without loss of generality, we can set whenever . Using the argument above, there are deterministic times such that (a.s.) whenever . Taking the limit as *n* goes to infinity, there exists a deterministic time *s* with

almost surely. Then, again,

A straightforward consequence of the measurable selection and section theorems is that, to test whether two processes are equal, it is enough to compare them at each random time, up to zero probability sets. Throughout, I am considering processes to be equal if they are equal up to evanescence.

Theorem 4LetXandYbe stochastic processes.

Measurable Selection: IfX,Yare jointly measurable and (a.s.) for each -measurable random time , then .Optional Section: IfX,Yare optional and (a.s.) for each stopping time , then .Predictable Section: IfX,Yare predictable and (a.s.) for each predictable stopping time , then .

*Proof:* Let . That is, *S* is the set of for which , and lies in the sigma-algebra generated by *X* and *Y*. Any random time with graph contained in *S* satisfies whenever . I will argue by contradiction, so suppose that . Then, *S* is not evanescent, hence .

If *X*,*Y* are jointly measurable then so is *S* and, by Theorem 1, there exists a random time with graph contained in *S* and,

contradicting the condition of the first statement.

Next, if *X*,*Y* are optional then so is *S* and, by Theorem 2, there exists a stopping time with graph contained in *S* and,

(3) |

Taking small enough that the right hand side is non-negative contradicts the condition of the second statement.

Finally, if *X*,*Y* are predictable then so is *S*. Theorem 3 states that there exists a predictable stopping time with graph contained in *S*, and such that (3) holds. This contradicts the condition of the third statement. ⬜

We previously showed that a map is measurable if and only if is jointly measurable, and is a stopping time if is progressive or, equivalently, is optional. In the previous post, it was also mentioned that is a predictable stopping time if and only if is a predictable set. However, advanced techniques are required to prove this statement, so no proof was given at the time. With the aid of the predictable section theorem, I will now give a proof.

Lemma 5A map is a predictable stopping time if and only if .

*Proof:* By predictable section, Theorem 3, there are predictable stopping times such that and

By the Borel-Cantelli lemma, for all large enough *n*, almost surely. Hence, is a predictable stopping time. ⬜

Compare this to the classification of predictable stopping times previously proven in these notes. There, a stopping time was shown to be predictable if and only if its graph is predictable. The proof used there relied on some advanced properties of the stochastic integral in order to show that implies that is *fair*. This required constructing the paths of stochastic integrals of with respect to bounded martingales. Although it requires predictable section, the proof given in this post is more direct, and is also much stronger, since it does not require to already be known to be a stopping time.

Not exactly a comment but an answer to your (George Lowther) request on RG, who don’t accept djvu (why?). You will find many of my books

(in djvu) on the russian library “genesis”. Here is one address (many mirrors): http://gen.lib.rus.ec/

Best regards

CD

Comment by Anonymous — 30 November 16 @ 11:20 AM |

Many thanks for that, I was able to find the book I was looking for (although my IP has actually blocked that website).

Incidentally, it’s nice to get a personal response from such a master and authority on the subject!

Regards

George Lowther

Comment by George Lowther — 30 November 16 @ 5:01 PM |

[…] this post I give a proof of the theorems of optional and predictable section. These are often considered among the more advanced results in stochastic calculus, and many texts […]

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[…] aim of this post is to give a fairly direct proof of the theorems of measurable projection and measurable section. These are generally regarded as rather difficult results, and proofs often use ideas from […]

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