5
$\begingroup$

I'm beginning to study about stochastic processes, and currently focusing on stopping times and hitting times. The textbook I'm using is "Stochastic Integration Theory" by Medvegyev (and Karatzas & Shreve as a second reference), and in some of the theorems the following measurable projection theorem is used.

If the space $(\Omega,\mathcal{A},\mathbb{P})$ is complete and $$U \in \mathcal{B}(\mathbb{R}^n) \otimes \mathcal{A},$$ then $$\text{proj}_\Omega (U) := \{x: \exists t\text{ such that }(t,x) \in U\} \in \mathcal{A}.$$

On the authors homepage there is a note containing a proof as well as many definitions such as Suslin (also called analytic) sets and auxiliary lemmas, however I find the material to be lacking in rigor and it is missing some assumptions. Therefore I am looking for a textbook in which the measurable projection is covered in detail. I've looked at the textbooks by Kechris, and Srivastava without finding what I was looking for.

$\endgroup$
  • $\begingroup$ Dellacherie and Meyer, Probabilities and Potential, Chapter III. $\endgroup$ – zhoraster Jan 8 '16 at 7:24
  • $\begingroup$ I didn't find this in Chapter III, but should certainly be inside this book. All such facts are there. $\endgroup$ – zhoraster Jan 8 '16 at 7:40
  • $\begingroup$ Yeah I just looked at Chapter III, but to be honest, the notation and organization is a little messy so it's hard to find the actual theorem. $\endgroup$ – Olorun Jan 8 '16 at 7:54
  • $\begingroup$ @Olorun Agreed with not best place... Anyways,,, suggestion. Take it or leave it. I had got the same helpful hint on Math.SE and Chemistry.SE both, so thought of passing it on... $\endgroup$ – Shailesh Jan 13 '16 at 4:40
4
$\begingroup$

Theorem 13 in Chapter III of the first volume of Dellacherie & Meyer (cited by @zhoraster; see the foot of page 43 in the English translation) tells you that the projection onto $\Omega$ of $U$ is $\mathcal A$-analytic. As such, this projection is $\mathcal A$-measurable, because $(\Omega,\mathcal A,\Bbb P)$ is complete; see no. III-33 at the top of p. 58 of D. & M.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.