The aim of this post is to give a 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 descriptive set theory such as analytic sets. I did previously post a proof along those lines on this blog. However, the results can be obtained in a more direct way, which is the purpose of this post. Here, I present relatively self-contained proofs which do not require knowledge of any advanced topics beyond basic probability theory.
The projection theorem states that if is a complete probability space, then the projection of a measurable subset of onto is measurable. To be precise, the condition is that S is in the product sigma-algebra , where denotes the Borel sets in , and the projection map is denoted
Then, measurable projection states that . Although it looks like a very basic property of measurable sets, maybe even obvious, measurable projection is a surprisingly difficult result to prove. In fact, the requirement that the probability space is complete is necessary and, if it is dropped, then need not be measurable. Counterexamples exist for commonly used measurable spaces such as and . This suggests that there is something deeper going on here than basic manipulations of measurable sets.
By definition, if then, for every , there exists a such that . The measurable section theorem — also known as measurable selection — says that this choice can be made in a measurable way. That is, if S is in then there is a measurable section,
It is convenient to extend to the whole of by setting outside of .
The graph of is
The condition that whenever can alternatively be expressed by stating that . This also ensures that is a subset of , and is a section of S on the whole of if and only if .
The results described here can also be used to prove the optional and predictable section theorems which, at first appearances, also seem to be quite basic statements. The section theorems are fundamental to the powerful and interesting theory of optional and predictable projection which is, consequently, generally considered to be a hard part of stochastic calculus. In fact, the projection and section theorems are really not that hard to prove.
Let us consider how one might try and approach a proof of the projection theorem. As with many statements regarding measurable sets, we could try and prove the result first for certain simple sets, and then generalise to measurable sets by use of the monotone class theorem or similar. For example, let denote the collection of all for which . It is straightforward to show that any finite union of sets of the form for and are in . If it could be shown that is closed under taking limits of increasing and decreasing sequences of sets, then the result would follow from the monotone class theorem. Increasing sequences are easily handled — if is a sequence of subsets of then from the definition of the projection map,
If for each n, this shows that the union is again in . Unfortunately, decreasing sequences are much more problematic. If for all then we would like to use something like
However, this identity does not hold in general. For example, consider the decreasing sequence . Then, for all n, but is empty, contradicting (1). There is some interesting history involved here. In a paper published in 1905, Henri Lebesgue claimed that the projection of a Borel subset of onto is itself measurable. This was based upon mistakenly applying (1). The error was spotted in around 1917 by Mikhail Suslin, who realised that the projection need not be Borel, and lead him to develop the theory of analytic sets.
Actually, there is at least one situation where (1) can be shown to hold. Suppose that for each , the slices
are compact. For each , the slices give a decreasing sequence of nonempty compact sets, so has nonempty intersection. So, letting S be the intersection , the slice is nonempty. Hence, , and (1) follows.
The starting point for our proof of the projection and section theorems is to consider certain special subsets of where the compactness argument, as just described, can be used. The notation is used to represent the collection of countable intersections, , of sets in .
Lemma 1 Let be a measurable space, and be the collection of subsets of which are finite unions over compact intervals and . Then, for any , we have , and the debut
is a measurable map with and .