Almost Sure

8 November 16

Measurable Projection and the Debut Theorem

I will discuss some of the immediate consequences of the following deceptively simple looking result.

Theorem 1 (Measurable Projection) If {(\Omega,\mathcal{F},{\mathbb P})} is a complete probability space and {A\in\mathcal{B}({\mathbb R})\otimes\mathcal{F}} then {\pi_\Omega(A)\in\mathcal{F}}.

The notation {\pi_B} is used to denote the projection from the cartesian product {A\times B} of sets A and B onto B. That is, {\pi_B((a,b)) = b}. As is standard, {\mathcal{B}({\mathbb R})} is the Borel sigma-algebra on the reals, and {\mathcal{A}\otimes\mathcal{B}} 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 {{\mathbb R}^2} onto {{\mathbb R}} to be Borel. In order to formalise this, we could start by noting that sets of the form {A\times B} 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 {S_n=(0,1/n)\times{\mathbb R}\subseteq{\mathbb R}^2} all project onto the whole of {{\mathbb R}}, 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 {\Omega={\mathbb R}} and {\mathcal{F}=\mathcal{B}({\mathbb R})}, 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 {(\Omega,\mathcal{F})} is assumed to be complete, and stochastic processes are taken to be real-valued, or take values in the extended reals {\bar{\mathbb R}={\mathbb R}\cup\{\pm\infty\}}, with time index ranging over {{\mathbb R}_+}. 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 {S\in\mathcal{B}(\mathbb{R}_+)} then {\sup_{s\in S}X_s} is measurable.

Proof: Setting {U=\sup_{s\in S}X_s} then, for each real K, {U > K} if and only if {X_s > K} for some {s\in S}. Hence,

\displaystyle  U^{-1}\left((K,\infty]\right)=\pi_\Omega\left((S\times\Omega)\cap X^{-1}\left((K,\infty]\right)\right).

By the measurable projection theorem, this is in {\mathcal{F}} and, as sets of the form {(K,\infty]} generate the Borel sigma-algebra on {\mathbb{\bar R}}, U is {\mathcal{F}}-measurable. ⬜

Next, the running maximum of a jointly measurable process is again jointly measurable.

Lemma 3 If X is a jointly measurable process then {X^*_t\equiv\sup_{s\le t}X_s} is also jointly measurable.

(more…)

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