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.
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.