compute the eigenvalue distribution $d\nu$. In our generalized scheme, it is given by formulae and . Note that the power of is reflected by the fact that it provides a direct and unified method to compute the eigenvalue distributions of various kinds of random matrix ensembles. In the sequel paper [@AWY], we will see that all the classical ensembles are included in the generalized scheme, and the corresponding eigenvalue distributions can be derived from and . We will also present various kinds of generalized ensemble which were not considered before, and compute their eigenvalue distributions explicitly.

0.3cm Now we consider the integration formula associated with the generalized random matrix ensemble. As in §3, we assume the following covering condition holds.

0.3cm (*covering condition*) The map $\varphi: G/K\times Y'\rightarrow X'$ is a $d$-sheeted covering map, with $d<+\infty$.

\[T:random-integration\] Let $(G,\sigma,X,p(x)dx,Y,dy)$ be a generalized random matrix ensemble. Suppose the covering condition (e) holds. Then we have the following integration formula $$\label{E:integration}
\int_X f(x)p(x)
dx=\frac{1}{d}\int_Y\left(\int_{G/K}f(\sigma_g(y))d\mu([g])\right)
d\nu(y)$$ for all $f\in C^\infty(X)$ with $f\geq0$ or with $f\in
L^1(X,p(x)dx)$. If moreover $f(\sigma_g(x))=f(x), \forall g\in G,
x\in X$, then $$\label{E:classintegration}
\int_X f(x)p(x) dx=\frac{\mu(G/K)}{d}\int_Y f(y)d\nu(y).$$

It is obvious by Theorem \[T:Weylbianhuan\] and Corollary \[C:leihanshubianhuan\].

In formula , if 