A stochastic process is a collection $(X_t)_{t\in T}$ of random variables from a prob. space $(\Omega,\mathcal{F},P)$ to some measurable space $(E,\mathcal{E})$. Now, in order to understand the whole process as a 'single' random variable one look at it as a map $X:\Omega\to E^T$, where $E^T=\prod_{t\in T}E$ is the set of all functions from $T$ to $E$.
$E^T$ is usually equipped with the product sigma algebra $\mathcal{E}^T$ which is produced by the one-dimensional cylinder sets, i.e. $\mathcal{E}^T=\sigma(\pi^{-^1}(A): A\in\mathcal{E}, t\in T)$ (is is the same as the sigma algebra produced by all finite-dimensional cylinders?), where $\pi_t$ is the projection map from $E^T$ to $E$. Now, if we are interested just in a subspace $S\subset E^T$, then it is often the case that $S\notin\mathcal{E}^T$.
For example take $T=[0,1]$, $E=\mathbb{R}$, $\mathcal{E}=\mathcal{B}_{\mathbb{R}}$ and $C[0,1]$ as a subspace of $\mathbb{R}^{[0,1]}$, then $C[0,1]\notin\mathcal{E}^T$.
By looking at $\mathcal{S}=S\cap \mathcal{E}^T=\{S\cap A:A\in\mathcal{E}^T\}$ one can avoid this problem. Now my main question: If our subspace $S$ is a metric space it is natural to look at the Borel $\sigma$-algebra $\mathcal{B}_S$. Is it always the case that $\mathcal{S}=\mathcal{B}_S$? I think that we need some additional property like separability, but I cannot prove it.
