# Integrating the inverse-Wishart density

It is alleged in this question and in the Wikipedia article and elsewhere that the density function for the inverse-Wishart distribution with $$n$$ degrees of freedom on $$p\times p$$ positive-definite symmetric matrices is $$f(x) = c\times (\det x)^{-(n+p+1)/2} \exp\left( \frac{-1}2 \operatorname{tr} \left(\Sigma x^{-1}\right) \right)$$ where $$\Sigma$$ is the scale matrix.

With respect to what measure is this the density? One could guess that it means $$\Pr( X\in A) = \int_A f(x) \prod_{i,j} dx_{ij}$$ where $$x_{ij}$$ is the $$ij$$ entry of the matrix $$x,$$ i.e. it's with respect to Lebesgue measure on a $$p^2$$-dimensional space. But then it's really a $$p(p-1)/2$$-dimensional space, since $$x$$ is symmetric, and that complicates the question. And that's something of a wild guess. The bounds of integration would be messy since the support of the density is the space of symmetric positive-definite matrices.