The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $d\in\mathbb{N}$ satisfy $1\le d$, and let $L^{2}(\Omega;\mathbb{R}^{d})$ be the space of classes of square-integrable random vectors in $\mathbb{R}^{d}$ on $(\Omega,\mathcal{F},P)$, with the inner product and norm $\langle\cdot,\cdot\rangle_{L^{2}}$, $\lVert\cdot\rVert_{L^{2}}$ and the notational convention fixed there. Then the following hold.

1. (Hilbert space)¶ $L^{2}(\Omega;\mathbb{R}^{d})$ is a real Hilbert space.

2. (Laws have finite second moment)¶ For every $X\in L^{2}(\Omega;\mathbb{R}^{d})$ the law $\mathcal{L}(X)$ belongs to the set $\mathcal{P}_{2}(\mathbb{R}^{d})$ of probability measures with finite second moment, with second moment $M_{2}(\mathcal{L}(X))=\lVert X\rVert_{L^{2}}^{2}$.

3. (Constants and translations)¶ For $a\in\mathbb{R}^{d}$ the constant map $\Omega\to\mathbb{R}^{d}$ with value $a$ is a square-integrable random vector, whose class, denoted $c_{a}$, satisfies $\lVert c_{a}\rVert_{L^{2}}=\lVert a\rVert$; and for $X\in L^{2}(\Omega;\mathbb{R}^{d})$ the law of $X+c_{a}$ is the push-forward of $\mathcal{L}(X)$ by the translation $\tau_{a}:\mathbb{R}^{d}\to\mathbb{R}^{d}$, $\tau_{a}(x)=x+a$, which is Borel.