Wishart law by Gaussian outer products

Assume the Axiom of Choice. Let $p,m\ge 1$ be integers and let $\mathrm{\Sigma }$ be a real symmetric positive-semidefinite $p\times p$ matrix, in the sense of Multivariate normal law, including singular covariance. Give ${\mathrm{S}\mathrm{y}\mathrm{m}}_p$, the space of real symmetric $p\times p$ matrices, its Euclidean Borel sigma-algebra by identifying a matrix with its upper-triangular entries. The finite product measure $N_p(0,\mathrm{\Sigma }{)}^{\otimes m}$ exists because each factor is a probability measure and hence sigma-finite. Define the Wishart law $W_p(\mathrm{\Sigma },m)$ as the pushforward of this product measure by the continuous map $(z_1,\dots ,z_m)\mapsto {\sum }_{j=1}^m{z}_j{z}_j^T$. Equivalently, it is the law of ${\sum }_{j=1}^m{Z}_j{Z}_j^T$ for independent $Z_j\sim N_p(0,\mathrm{\Sigma })$.

Here $\mathrm{\Sigma }$ is the scale matrix and the positive integer $m$ is the degrees of freedom. This definition allows a singular Wishart law when $m<p$ or $\mathrm{\Sigma }$ is singular; no Lebesgue density is included in the convention. The usual nonsingular density when $m\ge p$ is not needed below.