Let $\mathcal H\_n$ denote the space of Hermitian $n\times n$ matrices and let $\mu\_{GUE}$ denote the following measure on $\mathcal H\_n$:
$$
\mu\_{GUE}(dM) = \exp\left( -\frac{n}{2}\text{tr}\ M^2\right)\ dM,
$$
where $dM$ denotes the Lebesgue measure on $\mathcal H\_n$. For fixed $n$, $\mu\_{GUE}$ has finite mass. The resulting normalized measure is the GUE ensemble studied in Random Matrix Theory.


**Question:** Can this measure be extended to the space of self-adjoint operators?


**Fuzzy question:** Do people study Gaussian ensembles of random self-adjoint operators?



Extending GUE to a measure on operators?
One way of looking at this is that, up to normalization, the GUE is standard Gaussian measure on the space of Hermitian matrices, equipped with the Hilbert-Schmidt or Frobenius inner product. So what you're asking for is a special case of extending the notion of standard Gaussian measure to a particular infinite-dimensional space. But [you can't do that.](https://mathoverflow.net/a/36417/1044)


