I have a question about the application of the definition of a random measure on the empirical spectral distribution.
Let $(X , \mathcal{B})$ be some measurable space and let $(\Omega, \mathcal{F}, \mathbb{E})$ be a probability space. A random measure is a mapping $M : \Omega \times \mathcal{B} \to \mathbb{R}$ such that:
For each $\omega \in \Omega$, $M(\omega, \cdot)$ is a measure on $(X , \mathcal{B})$.
For each $A \in \mathcal{B}$, $M(\cdot, A)$ is a real-valued random variable.
I am having trouble in applying this definition of a random measure to the empirical spectral distribution.
Let a $n \times n$ Hermitian matrix $M_n$, we can form the (normalized) empirical spectral distribution
$$ \mu_{\frac{1}{\sqrt{n}} M_n} := \frac{1}{n} \sum_{j=1}^n \delta_{\lambda_j(M_n) / \sqrt{n}}, $$
of $M_n$, where $\lambda_1(M_n) \leq \ldots \leq \lambda_n(M_n)$ are the (necessarily real) eigenvalues of $M_n$, counting multiplicity. When $M_n$ is random then $\mu_{\frac{1}{\sqrt{n}} M_n}$ is a random measure.
If we fix $M_n$ is $\mu_{\frac{1}{\sqrt{n}} M_n}$ a measure on some measurable space $(X , \mathcal{B})$ or a real-valued random variable. And how would we define $(X , \mathcal{B})$ in this case ?