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.