The Gaussian Law with Given Mean and Covariance Matrix

In the setting of Probability Spaces: Notation, let $d\in\mathbb{N}$, $m\in\mathbb{R}^{d}$, and let $\Sigma\in\mathcal{S}(d)$ be positive semidefinite.

(i) Gaussian law. The Gaussian law (or normal distribution) with mean $m$ and covariance matrix $\Sigma$, written $N(m,\Sigma)$, is the law of any Gaussian random vector in $\mathbb{R}^{d}$, on any probability space, with mean $m$ and covariance matrix $\Sigma$ (a Gaussian random vector has a mean and a covariance matrix by Square-Integrability, Mean and Covariance Matrix of a Gaussian Random Vector (i)). Such a vector exists by The Law of a Gaussian Random Vector Is Determined by Its Mean and Covariance Matrix (iii), and its law does not depend on the choice of the vector by The Law of a Gaussian Random Vector Is Determined by Its Mean and Covariance Matrix (ii).