Let $K \subset \mathbb{R}$ be compact, and let $E$ be a separable Banach space. Further, let $(\Omega, \mathcal{F},\mathbb{P})$ a probability space. I would like to show that a certain a random function
$$
f \colon \to C(K,E)
$$
is measurable. This means I have to show $f^{-1}(B) \in \mathcal{F}$ for all $B \in \mathcal{B}(C(K,E))$, but I do not know where to start, as I do not know exactly how Borel sets $B$ in $\mathcal{B}(C(K,E))$. Can anyone help me getting started? 

Thanks