Standard real and normalized complex Gaussian matrix laws

Assume the Axiom of Countable Choice so that the repository's Lebesgue measure on each positive-dimensional real coordinate space is a measure. A standard real Gaussian scalar has Lebesgue density ${\phi }_R(x)=(2\pi {)}^{-1\mathrm{/}2}{e}^{-x^2\mathrm{/}2}\mathrm{.}$ A normalized complex Gaussian scalar is $z=(x+iy)\mathrm{/}\sqrt{2}$ for independent standard real Gaussian scalars $x,y$; its density on $\mathbb{C}\simeq {\mathbb{R}}^2$ is ${\phi }_C(z)={\pi }^{-1}{e}^{-\mathrm{\mid }z{\mathrm{\mid }}^2}$ and $\mathbb{E}\mathrm{\mid }z{\mathrm{\mid }}^2=1$.

For $F=\mathbb{R}$ or $\mathbb{C}$, an iid standard Gaussian matrix in $F^{a\times b}$ has independent entries with the corresponding scalar law. For $ab>0$, its expectations are finite-dimensional Lebesgue integrals against the product density. If $a=0$ or $b=0$, the matrix space has one element, the empty matrix; its law assigns that element probability one, and the expectation of any function is its value there. Independence means the product-density factorization, with this one-point convention for an empty product. The normalized complex convention is fixed throughout; it is not the unnormalized law $x+iy$.