Given two stochastic processes on a probability space, will their compound process be a valid stochastic process on the same probability space? Let the stochastic process $M=(M_t, t\ge 0)$ and the stochastic pathwise continuous increasing process $Y=(Y_t,t\ge 0)$ be defined on the probability space $(\Omega, \mathcal F, P)$. Will the compound process $M_Y=(M_{Y_t},t\ge 0)$ also be valid (measureable on the same sigma algebra which $M$ and $Y$ maps from) on this probability space? 
If it is not valid in general, what if $M$ and $v$ are independent of each other? Will it then be 'valid'?
Clarification:
Does 
$\{\omega\in\Omega\colon M(\omega)\in B\}\in\mathcal{F}$, $\forall B\in \mathcal{B}(\mathbb{R})$ and $\{\omega\in\Omega\colon Y(\omega)\in B\}\in\mathcal{F}$, $\forall B\in \mathcal{B}(\mathbb{R})$ $\implies$ $\{\omega\in\Omega\colon M_Y(\omega)\in B\}\in\mathcal{F}$, $\forall B\in \mathcal{B}(\mathbb{R})$
hold? $\mathcal{B}(\mathbb{R})$ being the generated Borel $\sigma$-algebra.

I guess that by "same sigma algebra which $M$ and $Y$ maps onto" you mean the sigma algebra generated by the two processes.
However, this is not the case. For instance take $Y$ to be the deterministic process defined by $Y_t=2t$ and $M_t$ any stochastic process such that its filtration satisfies that $\mathcal{F}_t\subsetneq\mathcal{F}_s$ for $t,s$.
At time $t$ the sigma algebra generated by $M$ and $Y$ is the same one as the one generated by $M$.
However, even $M_{Y_t}=M_{2t}$ is not measurable at time $t$.
