ensemble in an unified way in [@AWY].

0.3cm In §5 we will give a classification scheme of generalized ensembles, that is, the linear ensemble, the nonlinear noncompact ensemble, the compact ensemble, the group and algebra ensembles, as well as the pseudo-group and pseudo-algebra ensembles. According to this classification scheme, Gaussian ensemble and chiral ensemble are included in linear ensemble, new transfer matrix ensembles is included in nonlinear noncompact ensemble, circular and Jacobi ensembles are included in compact ensemble. Some new ensembles which were not considered before will also be included.

0.5cm

Geometry of $G$-spaces
======================

0.5cm In this section we develop some geometrical preliminaries which will be needed to establish our theory of the generalized random matrix ensembles.

0.3cm First we make some preparation about measures on manifolds. Let $M$ be an $n$-dimensional smooth manifold. A measure $dx$ on $M$ is called *smooth* (or *quasi-smooth*) if on any local coordinate chart $(U;
x_1,\cdots,x_n)$ of $M$, $dx$ has the form $dx=f(x)dx_1\cdots
dx_n$, where $f$ is a smooth function on $U$ and $f>0$ (or $f\geq
0$), $dx_1\cdots dx_n$ is the Lebesgue measure on $\mathbb{R}^n$. Note that the smooth measures on $M$ are unique up to multiplying a positive smooth function on $M$, 