and suppose $p(x)$ is a $G$-invariant smooth function on $X$. Then by Corollary \[C:J\], there is a quasi-smooth measure $d\nu$ on $Y$ such that $$\label{E:distribution}
\varphi^*(p(x)dx)=d\mu d\nu.$$

\[D:generalized\] Let the conditions and notations be as above. Then the system $(G,\sigma,X,p(x)dx,Y,dy)$ is called a *generalized random matrix ensemble*. The manifolds $X$ and $Y$ are called the *integration manifold* and the *eigenvalue manifold*, respectively. The measure $d\nu$ on $Y$ determined by is called the *generalized eigenvalue distribution*.

Recall that in §2 we have defined the map $\Psi_y : {\mathfrak{l}}\rightarrow T_yO_y$ by $$\Psi_y (\xi ) = \frac{d}{dt}\Big |_{t=0}\hskip 0.1cm \sigma_{\exp
t\xi}(y) , \quad \forall \xi\in {\mathfrak{l}},$$ where ${\mathfrak{l}}$ is a linear subspace of ${\mathfrak{g}}$ such that ${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{l}}$. Thanks to the preliminaries in §2, we can compute the generalized eigenvalue distribution directly according to the following theorem.

\[T:distribution\] Let $(G,\sigma,X,p(x)dx,Y,dy)$ be a generalized random matrix ensemble. Then the generalized eigenvalue distribution $d\nu$ is given by $$\label{E:distribution=}
d\nu(y)=\mathcal{P}(y)dy=p(y)J(y)dy,$$ where $$\label{E:factor=det}
J(y)= C |\det\Psi_y|,$$ here $C=|\det((d\pi)_e|_{\mathfrak{l}})|^{-1}$.

This follows directly from Proposition \[P:J\], Corollary \[C:J\], Proposition \[P:bian\] and Theorem \[T:det\].

The function $\mathcal{P}(y)=p(y)J(y)$ determined by formula is called the *generalized joint density function*.

0.3cm One of the most fundamental problems in the random matrix theory is to 