I have possibly a quite elementary question, which I'm assuming if the statement is not true there must exist several famous counter-examples. Given a Brownian motion $\Big( B_t \Big)_{t\in I}$, where $I=[0,a]$ or $I=[0,\infty)$, is it measurable in the product space $(\Omega\times I, \mathcal{F} \otimes \mathcal{B}_I)$?
For any $\omega\in \Omega$ the cross section with respect to $\omega$ of $B(t,\omega)$ is $B_\omega(t)$ which is $P$-a.s continuous and thus $P$-a.s measurable. And for any $t\in I$ the cross section with respect to $t$ of $B(t,\omega)$ is $B_t(\omega)$ which is $N(0,t)$ and also measurable. This seems to (for me at least) indicate strongly that it is product measurable? I would appreciate any answer or reference to helpful reading material.