Random Vector and Its Law

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$.

1. (Random vector)(i) A random vector in $\mathbb{R}^{d}$ on the probability space $(\Omega,\mathcal{F},P)$ is a map $X:\Omega\to\mathbb{R}^{d}$ that is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R}^{d})$. For $B\in\mathcal{B}(\mathbb{R}^{d})$ we write $\{X\in B\}$ for $X^{-1}(B)$, an event, and $P(X\in B)$ for its probability. By claim 3 of Euclidean Points as Tuples of Real Numbers, a map $X:\Omega\to\mathbb{R}^{d}$ is determined by its $d$ coordinates(ii), the maps $X_{i}:\Omega\to\mathbb{R}$ with value $X(\omega)_{i}$ at $\omega$ for $i\in[d]$, and any $d$ maps $\Omega\to\mathbb{R}$ are the coordinates of exactly one map $\Omega\to\mathbb{R}^{d}$; thus a random vector in $\mathbb{R}^{d}$ may be written as a $d$-tuple $(X_{1},\dots,X_{d})$ of maps, as in Gaussian Random Vectors and Jointly Gaussian Random Variables.

2. (Law)(iii) The law of a random vector $X$ in $\mathbb{R}^{d}$ is the image measure of $P$ under $X$, denoted $\mathcal{L}(X)$:

$$\mathcal{L}(X)(B)=P(X\in B)\qquad(B\in\mathcal{B}(\mathbb{R}^{d})).$$

By claim 1 of that lemma, $\mathcal{L}(X)$ is a probability measure on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$, that is, $\mathcal{L}(X)\in\mathcal{P}(\mathbb{R}^{d})$. For $\mu\in\mathcal{P}(\mathbb{R}^{d})$ we write $X\sim\mu$ when $\mathcal{L}(X)=\mu$, and say that $X$ has law $\mu$. This is the construction of the distribution $P_{V}$ of a random variable $V$, which is the image measure of $P$ under $V$ on $\mathcal{B}(\mathbb{R})$, carried over to maps with values in $\mathbb{R}^{d}$.