I will discuss some of the immediate consequences of the following deceptively simple looking result.
Theorem 1 (Measurable Projection) If is a complete probability space and then .
The notation is used to denote the projection from the cartesian product of sets A and B onto B. That is, . As is standard, is the Borel sigma-algebra on the reals, and denotes the product of sigma-algebras.
Theorem 1 seems almost obvious. Projection is a very simple map and we may well expect the projection of, say, a Borel subset of onto to be Borel. In order to formalise this, we could start by noting that sets of the form for Borel A and B have an easily described, and measurable, projection, and the Borel sigma-algebra is the closure of the collection such sets under countable unions and under intersections of decreasing sequences of sets. Furthermore, the projection operator commutes with taking the union of sequences of sets. Unfortunately, this method of proof falls down when looking at the limit of decreasing sequences of sets, which does not commute with projection. For example, the decreasing sequence of sets all project onto the whole of , but their limit is empty and has empty projection.
There is an interesting history behind Theorem 1, as mentioned by Gerald Edgar on MathOverflow (1) in answer to The most interesting mathematics mistake? In a 1905 paper, Henri Lebesgue asserted that the projection of a Borel subset of the plane onto the line is again a Borel set (Lebesgue, (3), pp 191–192). This was based on the erroneous assumption that projection commutes with the limit of a decreasing sequence of sets. The mistake was spotted, in 1916, by Mikhail Suslin, and led to his investigation of analytic sets and to begin the study of what is now known as descriptive set theory. See Kanamori, (2), for more details. In fact, as was shown by Suslin, projections of Borel sets need not be Borel. So, by considering the case where and , Theorem 1 is false if the completeness assumption is dropped. I will give a proof of Theorem 1 but, as it is a bit involved, this is left for a later post.
For now, I will state some consequences of the measurable projection theorem which are important to the theory of continuous-time stochastic processes, starting with the following. Throughout this post, the underlying probability space is assumed to be complete, and stochastic processes are taken to be real-valued, or take values in the extended reals , with time index ranging over . For a first application of measurable projection, it allows us to show that the supremum of a jointly measurable processes is measurable.
Lemma 2 If X is a jointly measurable process and then is measurable.
Proof: Setting then, for each real K, if and only if for some . Hence,
By the measurable projection theorem, this is in and, as sets of the form generate the Borel sigma-algebra on , U is -measurable. ⬜
Next, the running maximum of a jointly measurable process is again jointly measurable.
Lemma 3 If X is a jointly measurable process then is also jointly measurable.
Lemma 4 If X is a progressively measurable process and for some time t, then is -measurable.
Proof: By definition of progressive measurability, for each time t, is -measurable. Hence, by Lemma 2,
is -measurable. ⬜
We can also show that the running maximum of a progressively measurable process is itself progressive.
Lemma 5 If X is an a progressively measurable process then is itself progressively measurable so, in particular, is adapted.
Proof: By definition, for any fixed time , the process is -measurable. Lemma 3 then says that is also -measurable, so is progressive. ⬜
It is often useful to look at the left or right limits of stochastic processes. As arbitrary processes do not necessarily have paths for which these limits exist, we look at the limit supremum. The limit supremum and left/strict-left/right/strict-right limits of a process are defined as
To be clear, when I take for in these definitions so that, for example, . Measurable projection enables us to show that these processes are progressively measurable processes whenever X is.
Lemma 6 If X is a progressively measurable process then,
are progressively measurable. If, furthermore, the filtration is right-continuous, then
are progressively measurable.
Proof: For each positive integer n, choose a sequence of times increasing to infinity, and such that the mesh goes to zero as n goes to infinity. For example, . Define the process by and
is progressive. This also implies that
is progressive. Now, with times as above, define the processes by
for t in the interval . This is right-continuous and is equal to . By Lemma 4, is -measurable over . Choosing any times , for all large enough n there exists k with . So, will be -measurable. Letting n go to infinity and then letting s increase to t shows that is -measurable. Furthermore, is -measurable for all positive , so is -measurable. Then,
is -measurable. If for all t this shows that Z is progressive. Then,
are also progressive. ⬜
The graph of a random time is
Measurable projection allows us to classify stopping times by the measurability of their graphs.
Lemma 7 Let be any map. Then,
- is measurable if and only if is jointly measurable.
- is a stopping time if and only if is progressive.
Proof: Clearly, if is measurable then will be jointly measurable and, if is a stopping time then
is a difference of right and left-continuous adapted processes and, hence, is progressive.
Conversely, setting then,
The Debut Theorem
A common use of measurable projection is in the proof of the debut theorem. This states that the first time that a progressively measurable process enters a (Borel) measurable set is a stopping time. We have already proven a version of this for right-continuous processes in these notes, which only required elementary measure theory. I now look at the general version.
The debut of a set is
The debut theorem states that this is a stopping time when A is progressive.
Theorem 8 (Debut Theorem) If is progressively measurable, and the underlying filtration is right-continuous, then is a stopping time.
Proof: For each positive time t, the set of for which is
The debut theorem implies that, for progressive processes, hitting times of measurable sets are stopping times. When X is right-continuous and K is closed, this was proven by more elementary methods much earlier in these notes. The generalisation here requires measurable projection.
Theorem 9 If X is a progressively measurable process and is Borel then, assuming the underlying filtration is right-continuous,
is a stopping time.
Proof: This follows immediately from Theorem 8, as is the debut of the progressively measurable set . ⬜
- Edgar, G. (2014) Most interesting mathematics mistake? MathOverflow. Link.
- Kanamori, A. (1995) The Emergence of Descriptive Set Theory. From Dedekind to Gödel: Essays on the Deveopment of the Foundations of Mathematics, Sythèse Library volume 251, 241–262. doi:10.1007/978-94-015-8478-4_10. Free pdf available from his website.
- Lebesgue, H. (1905) Sur les fonctions représentables analytiquement. Journal de Mathématiques Pures et Appliquées. Vol. 1, 139–216. Link.