= Solution
The spectral characterization of the <Reproducing-kernel Hilbert space of a stationary Gaussian process> says that its elements are precisely the functions whose Fourier transforms satisfy
$$
\lVert f\rVert_{\mathcal H_K}^2
=c\int_{\mathbb R}\frac{|\widehat f(u)|^2}{\widehat K(u)}\,du<\infty,
$$
with $c$ determined only by the <Fourier transform> convention. Here $\widehat K(u)=(1+u^2)^{-1}$, so
$$
\lVert f\rVert_{\mathcal H_K}^2
=c\int_{\mathbb R}(1+u^2)|\widehat f(u)|^2\,du.
$$
This is an equivalent norm for the <Sobolev space> $H^1(\mathbb R)$, and hence the RKHS equals $H^1(\mathbb R)$ as a set.
Since $\widehat K$ is integrable, $K$ is continuous and the process has a jointly measurable separable version. For every finite Borel measure $\nu$ on $\mathbb R$, <Tonelli theorem> gives
$$
\mathbb E\lVert X\rVert_{L^2(\nu)}^2
=\int_{\mathbb R}\mathbb E|X(t)|^2\,d\nu(t)
=K(0)\nu(\mathbb R)<\infty.
$$
Thus $X\in L^2(\mathbb R,\nu)$ almost surely and is a Borel random variable there. Every continuous linear functional of $X$ is a centered normal random variable: approximate its $L^2(\nu)$ integral by finite linear combinations of process values and pass to the $L^2$ limit. Therefore the induced law is a <Gaussian Borel measure> on $L^2(\mathbb R,\nu)$.
Solved by gpt-5.6-sol high.
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