Proof of the measurability of an integral process

Question

Given an square-integrable and adapted process $\Phi(s,\omega)$ with respect to a filtration $(\mathscr F_t)_{t>0}$ such that $P(\int_0^T|\Phi(s,\omega)|^2ds<\infty)=0$ . How can I prove that $$\int_0^q|\Phi(s,\omega)|^2ds$$ is adapted to the filtration $\mathscr F_q$.

Thanks

Answer 1

It is not enough that $\Phi$ be aan adapted process. You need to know that $\Phi$ is progressively measurable, which means that for each fixed $q>0$ the map $\Omega\times[0,q]\ni(\omega,t)\mapsto \Phi_t(\omega)$ is $\mathcal F_q\otimes\mathcal B([0,q])$-measurable. This permits the application of Tonelli's theorem.