A classic problem in this connection is to derive GOE or GUE statistics for the spectrum of a random self-adjoint operator of the form $H=-\nabla^2 + V(\vec{r})$, in some bounded domain of $\mathbb{R}^3$. The measure is the Gaussian measure for $V(\vec{r})$, of zero mean and given two-point correlation function. Since this is a real operator, one would expect GOE statistics, to obtain GUE statistics one would replace $\nabla\mapsto \nabla+i\vec{B}\times\vec{r}$ for some given vector $\vec{B}$.
This problem was solved by Konstantin Efetov in 1982, as described in much detail in his book on Supersymmetry in Disorder and Chaos. The GOE or GUE statistics is found to hold only over a limited range $E_{0}$ (the so-called Thouless energy): eigenvalues that differ by more than $E_0$ become uncorrelated, reverting to Poisson statistics.