This is mostly a reference request. I have integrals of the type \begin{equation} \int_C f(A) (dA) \end{equation} where $f$ is a realvalued function of a positive(semi)definite matrix argument, and the integration region $C$ is an interval in the cone of positivedefinite matrices, such as $C=[0.I]$, where this cone interval denotes the set of all positivedefinite matrices with positive eigenvalues all less than one. Other cone intervals could also occur, but in most cases they can be transformed to this or a similar form. Mostly the unctions $f$ will be symmetric functions in the sense that $f(AB)=f(BA)$, where $A$ and $B$ are positivedefinite matrices.There must be some papers about this kind of problem?
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I think that you could transform the problem through the use of tools from Chapter 5 in the following book There are explicit examples there for integration over spaces of symmetric positive definite matrices. 

