In a previous class we covered Gaussian processes and specified them by their mean functions and covariance kernels.
Is there a reference book that covers the construction of the corresponding Gaussian measure on the separable Banach space $C(K)$ where $K$ is a compact subset of $\mathbb{R}^{n}$ (or other appropriate space)?
I think the closest reference is Gaussian Measures in Hilbert Space by Kukush.
In this text the author first constructs a standard Gaussian measure on
$\mathbb{R}^{\mathbb{N}}$. This construction relies on Kolmogorov's extension theorem.
With a standard Guassian measure on $\mathbb{R}^{\mathbb{N}}$ in hand, it is then shown that under some mild assumption, $l_2(\mathbb{N})$ is a Borel subset of $\mathbb{R}^{\mathbb{N}}$ whose measure is one. Restricting then gives a standard Gaussian on $l_2(\mathbb{N})$.
With this construction in mind, since separable Hilbert spaces are isomorphic to $l_2(\mathbb{N})$, we can translate between covariance kernels $k(\cdot, \cdot)$ on a separable RKHS and positive-semidefinite operators $\Sigma$ on $l_2(\mathbb{N})$ via Mercer's theorem in the obvious way.
Once we have a positive-semidefinite operator $\Sigma$ on $l_2(\mathbb{N})$, we can push our standard Gaussian forward via an appropriate linear operator to obtain a Gaussian measure whose covariance operator is $\Sigma$ in way analogous to how it is done in finitely many dimensions.
Then we push this Gaussian measure back to the RKHS via the isomorphism to arrive at a Gaussian measure on our RKHS whose covariance operator is the integral operator induced by the covariance kernel.