I often encounter integrals over the space of real symmetric positive definite matrices, $$ \begin{equation} \int_{\mathbf{X>0}} f(\mathbf{X}) d\mathbf{X}, \end{equation} $$ where $f$ is a scalar function. One example is the multivariate gamma function.
In basic terms, how are these integrals defined?
From the reference in @光復香港時代革命FreeHongKong's comment it seems that
$$ \begin{equation} \int_{\mathbf{X>0}} f(\mathbf{X}) d\mathbf{X} = \int_{\mathbf{\mathbf{L}^+}} f(\mathbf{L} \mathbf{L}') J_{\mathbf{L}\mathbf{L}'}(\mathbf{L}) \, d\mathbf{L} = \int_{\mathbf{\mathbf{L}^+}} f(\mathbf{L} \mathbf{L}') 2^p \prod_{i=1}^{p} \mathbf{L}_{ii}^{p-i+1} \, d\mathbf{L} \end{equation}, $$ where $\mathbf{L}^+$ denotes the space of all lower triangular matrices with positive diagonal elements and $J_{\mathbf{L}\mathbf{L}'}(\mathbf{L})=2^p \prod_{i=1}^{p} \mathbf{L}_{ii}^{p-i+1}$ is the Jacobian of the transformation $\mathbf{L} \rightarrow \mathbf{L} \mathbf{L}'$.
Now, integrating over $\mathbf{L}^+$ boils down to a multiple integral over the lower triangluar elements, which for the elements below the main diagonal runs from $-\infty$ to $\infty$ and for the elements on the main diagonal runs from $0$ to $\infty$.