Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality

In the setting of Probability Spaces: Notation, let $m,n\in\mathbb{N}$.

(i) Coordinates. A function $X:\Omega\to\mathbb{R}^{n}$ is a random vector if and only if each of its coordinates $X_{1},\dots,X_{n}$ is a random variable.

(ii) Functions of random elements. Let $(S,d)$ and $(T,e)$ be metric spaces with Borel $\sigma$-algebras $\mathcal{B}(S)$ and $\mathcal{B}(T)$, let $X$ be a random element of $S$, and let $g:S\to T$ be $\mathcal{B}(S)/\mathcal{B}(T)$-measurable. Then $g(X)$ is a random element of $T$, and its law is the image measure of $P_{X}$ under $g$:

$$P_{g(X)}=P_{X}\circ g^{-1}.$$

In particular, if $X$ is a random vector in $\mathbb{R}^{n}$ and $g:\mathbb{R}^{n}\to\mathbb{R}^{m}$ is measurable, then $g(X)$ is a random vector in $\mathbb{R}^{m}$.

(iii) Pairs. Let $X$ be a random vector in $\mathbb{R}^{m}$ and $Y$ a random vector in $\mathbb{R}^{n}$. Then the function

$$(X,Y):\Omega\to\mathbb{R}^{m+n},\qquad\omega\mapsto\iota\bigl(X(\omega),Y(\omega)\bigr),$$

with $\iota$ the concatenation function, is a random vector in $\mathbb{R}^{m+n}$.

(iv) Almost-sure equality. Let $X$ and $Y$ be random vectors in $\mathbb{R}^{n}$. Then $\{X\neq Y\}$ is an event, and if $X=Y$ almost surely, then $X$ and $Y$ have the same law:

$$P_{X}=P_{Y}.$$