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.
2 Answers
I think you can adapt the proof in these notes of mine, 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.
You can definitely have Gaussian measures on Frechet spaces. However see Bogachev's Gaussian Measures Theorem 3.6.5 to see that you always have an embedded full measure Banach space.
