I'm considering Gaussian process on open domain $T$ in $\mathbb{R}^n$ and I tried to follow the abstract Wiener space construction of Gross. Since my sample paths are meant to be continuous, I thought that $sup$ norm would be nice measurable norm to complete the RKHS of mine (just as classical Wiener measure on $C[0,1]$), but since $T$ is open, this is not a good choice. Instead I considered countable family of semi norms $sup\_{K\_j}, j=1,2,...$ where compact sets $K\_j$ exhaust $T$. However with this family of semi norms $C(T)$ is merely Frechet space, not Banach space, hence I cannot use Gross's construction. Hence I like to know is there a generalized version of Gorss's construction, for example where measurable norms are replaced by family of semi norms, Banach space $W$ is replaced by Frechet space etc. One thing I noticed is that on "Encyclopedia of Mathematics" page about Wiener measure, the comment mentions about Wiener measure defined on $C[0,\infty)$, which seems to be closely related to my question, but I couldn't find the reference about it.



Reference for Wiener type measure on $C(T)$ when $T$ is open
I think you can adapt the proof in [these notes of mine](https://arxiv.org/abs/1607.03591v2), Theorem 4.44.


The first step in the proof of my notes is to pick, for each $n$, a finite-rank projection $P\_n$ such that for all finite-rank projections $P \perp P\_n$, we have
$$\tilde{\mu}\left(\left\{ h \in H : \|Ph\|\_W > 2^{-n}\right\}\right) < 2^{-n}.$$
Suppose instead we have a countable sequence of measurable seminorms $|\cdot|\_k$ $k=1,2,3,\dots$, which define the topology of $W$. Then for each $n$, we can apply this argument $n$ times (enlarging the projection $P\_n$ as we go) to instead find $P\_n$ such that for $P \perp P\_n$ we have
$$\tilde{\mu}\left(\left\{ h \in H : |Ph|\_k > 2^{-n}\right\}\right) < 2^{-n} , \qquad k = 1, 2, \dots, n.$$
A few steps later, we then obtain for each $k$ that
$\mathbb{P}(|S\_n - S\_{n-1}|\_k > 2^{-n}) < 2^{-n}$ for all $n \ge k$. This again implies that $\sum\_n \mathbb{P}(|S\_n - S\_{n-1}|\_k > 2^{-n}) < \infty$. Borel-Cantelli gives us that for each $k$, $\mathbb{P}$-a.s., $S\_n$ is Cauchy in $|\cdot|\_k$.


Taking a countable intersection, we in fact have that, $\mathbb{P}$-a.s., $S\_n$ is Cauchy in $|\cdot|\_k$ for all $k$. That is, on some event $E$ with $\mathbb{P}(E)=1$, we have for all $\omega \in E$ that $S\_n(\omega)$ is Cauchy in $|\cdot|\_k$ for all $k$. Since our space is complete, this implies that $S\_n(\omega)$ converges in $W$ to some $S(\omega)$. By standard arguments, $S$ is measurable, and its law $\mu = \mathbb{P} \circ S^{-1}$ is the measure we want.


