Let (\Omega,\mathcal{F},P) be a probability space. Let (X(t))_{t\geq 0} be a stochastic process (so the time domain is [0,\infty)).

Suppose that the stochastic process considered as a map [0,\infty)\times\Omega\rightarrow\mathbb{R} given by (t,\omega)\mapsto X(t,\omega) is measurable with respect to the product sigma-algebra \mathcal{B}([0,\infty))\otimes\mathcal{F}. Why then does it follow that for fixed \omega\in\Omega, the function t\mapsto X(t) is a Borel function?

I'm thinking that the restriction of a measurable function to a measurable set is measurable, but the point set \{\omega\} need not be measurable.