Gaussian Processes (GP) are widely viewed as practical ways to implement Gaussian Measures (GM) numerically. In fact, in many contexts it seems that to each GP corresponds a GM and vice versa, see, e.g., this. The focus there is mainly on GP/M on infinite dimensional spaces. In particular, they consider the separable Frechet spaces $C(I),\,C^k(I),\,AC(I)$, where $I$ is a real interval. Separability is an important assumption that ensures the fact that
The Borel sigma algebra on a Frechet space $X$ coincides with the smallest sigma algebra which makes continous linear functionals on $X$ measurable (see top p.3 here)
Replacing the real interval with a bounded open set in $\mathbb R^n$ changes nothing in their approach. My question is whether the result holds also for $C^\infty$, let's say on the unit cube -- open or closed. I could not adjust the proof for $C^k$ (obviously), nor could I find a reference, despite scouring the internet.
Does anyone know the answer for the question in the title? Thank you!