Let $X: (\mathbb R_+ \times \Omega, \mathcal B(\mathbb R_+) \otimes \mathcal F ) \rightarrow (\mathbb R_+, \mathcal B(\mathbb R_+))$ be a jointly measurable map. I would like to prove that the restriction of $X$ to the space $[0,T]\times \Omega$ is $\mathcal B[0,T] \otimes \mathcal F$ measurable as well. Is this true?
What I trie: Start with defining
$$ Y: ([0,T]\times \Omega, \mathcal B[0,T] \otimes \mathcal F) \rightarrow (\mathbb R_+, \mathcal B(\mathbb R_+)): \qquad Y(t,w) = X(t,w),\quad \forall (t,\omega) \in [0,T]\times \Omega $$
and to try to prove $Y^{-1}(-\infty,b]$ is measurable. What I see is
$$ Y^{-1}(-\infty,b] = \{(t,\omega) \in [0,T]\times \Omega : X(t,w) \leq b \} = \Big([0,T]\times \Omega\Big) \cap X^{-1}(-\infty,b] $$
and that $X^{-1}(-\infty,b] \in \mathcal B(\mathbb R_+) \otimes \mathcal F$. But how do we conclude that $Y^{-1}(-\infty,b] \in \mathcal B[0,T] \otimes \mathcal F$. Thank you for your help!