Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d,m\in\mathbb{N}$ satisfy $1\le d$ and $1\le m$, and let $X$ and $Y$ be random vectors in $\mathbb{R}^{d}$ on $(\Omega,\mathcal{F},P)$, with coordinates $X_{i}$, $Y_{i}$ and laws $\mathcal{L}(X)$, $\mathcal{L}(Y)$. For $a\in\mathbb{R}$ the maps $X+Y$, $aX$ and $X-Y$ are defined pointwise by the vector operations of $\mathbb{R}^{d}$. Integrability of a random variable is integrability with respect to $P$, and $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ are the coordinate projections for the splitting $d+d$, as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs. Then the following hold.

1. (Coordinates)¶ A map $Z:\Omega\to\mathbb{R}^{d}$ is a random vector in $\mathbb{R}^{d}$ if and only if each of its coordinates $Z_{1},\dots,Z_{d}$ is a random variable.

2. (Borel images and arithmetic)¶ If $g:\mathbb{R}^{d}\to\mathbb{R}^{m}$ is Borel, then $g\circ X$ is a random vector in $\mathbb{R}^{m}$ with law $\mathcal{L}(g\circ X)=g_{\#}\mathcal{L}(X)$. If $\varphi:\mathbb{R}^{d}\to\mathbb{R}$ is Borel, then $\varphi\circ X$ is a random variable, and if $\varphi:\mathbb{R}^{d}\to[0,\infty]$ is Borel, then $\varphi\circ X$ is measurable in the sense of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable. The maps $X+Y$, $aX$ and $X-Y$ are random vectors in $\mathbb{R}^{d}$, and the maps $\omega\mapsto\lVert X(\omega)\rVert$, $\omega\mapsto\lVert X(\omega)\rVert^{2}$, $\omega\mapsto X(\omega)\cdot Y(\omega)$ and $\omega\mapsto\lVert X(\omega)-Y(\omega)\rVert^{2}$ are random variables, denoted $\lVert X\rVert$, $\lVert X\rVert^{2}$, $X\cdot Y$ and $\lVert X-Y\rVert^{2}$.

3. (Change of variables)¶ For every Borel $\varphi:\mathbb{R}^{d}\to[0,\infty]$,

with $\mathbb{E}$ read as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §probability-space; and a Borel $\varphi:\mathbb{R}^{d}\to\mathbb{R}$ is integrable with respect to $\mathcal{L}(X)$ if and only if $\varphi\circ X$ is integrable, in which case the displayed identity holds in $\mathbb{R}$.

4. (Almost sure equality)¶ The set $\{X=Y\}=\{\omega\in\Omega:X(\omega)=Y(\omega)\}$ is an event, whose probability is written $P(X=Y)$; and if $P(X=Y)=1$, then $\mathcal{L}(X)=\mathcal{L}(Y)$.

5. (Pairs)¶ The pairing $(X,Y):\Omega\to\mathbb{R}^{d+d}$ is a random vector in $\mathbb{R}^{d+d}$, and its law is a coupling of $\mathcal{L}(X)$ and $\mathcal{L}(Y)$ with quadratic cost $I(\mathcal{L}((X,Y)))=\mathbb{E}[\lVert X-Y\rVert^{2}]$. Conversely, every random vector $Z$ in $\mathbb{R}^{d+d}$ equals $(\mathrm{pr}_{1}\circ Z,\mathrm{pr}_{2}\circ Z)$, the pairing of two random vectors in $\mathbb{R}^{d}$.