Recently i'm studying the book of Stroock and Varadhan - "Multidimensional Diffusion Processes" and i'm trying to solve this exercise : 
For the first part i argue in the following way :
"Because M is Polish there is a sequence of $\mathscr{F}$-measurable simple functions $\{f_{k}\}_{k\in\mathbb{N}}$, such that $f_{k}\to f$ pointwise. By the last result we can reduct the problem to f simple and after to the case $f=\chi_{A_{0}}$ with $A_{0}\in \mathscr{F}$.
Now, for each countable $I\subset [0,\infty]$ denote by $\mathscr{B}_{X^I}$ the product Borel $\sigma$-algebra in $X^{I}$, and define the $\mathscr{F}/\mathscr{B}_{X^I}$-measurable map $\rho_{I}:E\to X^{I}$ given by $$\rho_{I}(q)=\left(\eta(t,q)\right)_{t\in I}.$$ Now, note that
$$\Lambda = \left\{A\in\mathscr{F}; A=\rho_{I}^{-1}(B_{I})\, \mbox{for some countable}\, I\subset (0,\infty]\,\mbox{ and}\, B_{I}\in \mathscr{B}_{X^I}\right\}\subseteq \mathscr{F}$$
it's an $\sigma$-algebra and contains the set the generators $\mathscr{F}$, i.e. $\Lambda=\mathscr{F}$, therefore exists a countable $I_{0}\subset [0,\infty)$ and $B_{I_{0}}\in \mathscr{B}_{X^{I}}$ such that $A_{0}=\rho_{I_{0}}^{-1}(B_{I_{0}})$, and therefore
$$\chi_{A_{0}}=\chi_{\rho_{I_{0}}^{-1}(B_{I_{0}})}(q)=\chi_{B_{I_{0}}}\circ \rho_{I_{0}}(q)=\chi_{B_{I_{0}}}\left((\eta(t,q))_{t\in I}\right).$$
and the first part follows."
For the second part i really can't easy adapt the last argue to gets the result except on the case that $\theta$ is $\sigma(\eta)/\mathscr{B}_{M}$-measurable, cause in this case i use the fact that every adapted and right-continuous processes are progressively measurable and i apply this to $\eta$. I apreciate every suggests to solve this second part.
Another question about the result is : We can make the choose of $\{t_{i}\}_{i\in \mathbb{N}}$ independent of $t$?. (Because my choose of this sequence for the case $\theta$ being $\sigma(\eta)/\mathscr{B}_{M}$-measurable depends of t)
Thanks for any help.
PS: The answer for this exercise it's a good answer for this question from MSE in the case Y jointly measurable.