the measure $p(x)dx$ is a probability measure, and we let $f=1$, we get $\frac{\mu(G/K)}{d}\int_Y d\nu(y)=1$. So if $G/K$ is compact, we can normalized the measure $d\mu$ such that $\mu(G/K)=d$, then the generalized eigenvalue distribution $d\nu$ is a probability measure.

The condition $f\in C^\infty(X)$ in Theorem \[T:random-integration\] is superfluous. In fact, it is sufficient to assume $f$ is measurable. The same is true for Proposition \[P:jifenbianhuan\] and Theorem \[T:Weylbianhuan\].

0.5cm

A classification scheme of generalized ensembles
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0.5cm In this section we give a classification scheme of the generalized random matrix ensembles, that is, 0.2cm [(1)]{} Linear ensemble,\
(2) Nonlinear noncompact ensemble,\
(3) Compact ensemble,\
(4) Group ensemble,\
(5) Algebra ensembles,\
(6) Pseudo-group ensemble,\
(7) Pseudo-algebra ensemble.

0.3cm First we define the linear ensemble and the nonlinear noncompact ensemble. Let $G$ be a real reductive Lie group with Lie algebra ${\mathfrak{g}}$ in the sense of Knapp [@Kn], Section 7.2. Then $G$ admits a global Cartan involution $\Theta$, which induces a Cartan involution $\theta$ of ${\mathfrak{g}}$. Let the corresponding Cartan decomposition of ${\mathfrak{g}}$ is ${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{p}}$. Let $K=\{g\in
G:\Theta(g)=g\}$, $P=\exp({\mathfrak{p}})$, then $K$ is a maximal compact subgroup of $G$ with Lie algebra ${\mathfrak{k}}$, $P$ is a closed submanifold of $G$ satisfies $T_eP={\mathfrak{p}}$. The 