geometry of $G$-space which will be required to establish the generalized ensemble. After presenting four conditions, that is, the invariance condition, the transversality condition, the dimension condition, and the orthogonality condition, on which the definition of generalized ensemble will be based, we will prove in Theorem \[T:det\] a primary form of Formula .

0.3cm §3 will be devoted to the integration over $G$-spaces, which will be needed when we derive the integration formula associated with the generalized random matrix ensemble. Based on the four conditions presented in §2 and a covering condition, we will prove an integration formula in Theorem \[T:Weylbianhuan\], which converts the integration over a $G$-space to the integration by first integrating over each $G$-orbit, and then integrating over the orbits space. Two criterions on when the covering condition holds will also be given.

0.3cm Prepared by the preliminaries of §2 and §3, In §4 we will give the precise definition of the generalized random matrix ensemble, as well as the associated generalized eigenvalue distribution and generalized joint density function. In Theorem \[T:distribution\] the Formula will be presented, from which the associated eigenvalue distribution measure and density function will be derived for various concrete examples of the generalized 