We will shortly give a proof of measurable projection and, also, of the section theorems. Starting with the projection theorem, recall that this states that if is a complete probability space, then the projection of any 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 is the projection . Then, . 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.

The techniques which will be used to prove the projection theorem involve analytic sets, which will be introduced in this post, with the proof of measurable projection to follow in the next post. [**Note:** I have since posted a more direct proof of measurable projection and section, which does not make use of analytic sets.] These results can also be used to prove the optional and predictable section theorems which, at first appearances, 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, although the method given here does require stepping outside of the usual setup used in probability and involves something more like descriptive set theory. (more…)