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

Each result cited is universally quantified over the data in its own statement. By The Space of Square-Integrable Random Vectors §inner-product, $L^{2}(\Omega;\mathbb{R}^{d})$ is a real inner product space with norm $\lVert[X]\rVert_{L^{2}}=\sqrt{\mathbb{E}[\lVert X\rVert^{2}]}$ and distance $\delta([X],[Y])=\lVert[X]-[Y]\rVert_{L^{2}}$; here $[X]-[Y]=[X]+[(-1)Y]=[X+(-1)Y]=[X-Y]$, since the additive inverse of $[Y]$ is $[(-1)Y]$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §vector-space, by The Space of Square-Integrable Random Vectors §classes, and by claims 2 and 3 of Euclidean Space $\mathbb{R}^n$ is a Real Vector Space; so $\delta([X],[Y])=\sqrt{\mathbb{E}[\lVert X-Y\rVert^{2}]}$. By Real Hilbert Space §topology, Cauchy sequences and convergence in $L^{2}(\Omega;\mathbb{R}^{d})$ refer to the metric space $(L^{2}(\Omega;\mathbb{R}^{d}),\delta)$, and by Real Hilbert Space §hilbert claim 1 asserts that this metric space is complete: every Cauchy sequence converges.

Claim 1. Let $([X^{n}])_{n\in\mathbb{N}}$ be a Cauchy sequence in $(L^{2}(\Omega;\mathbb{R}^{d}),\delta)$, with representatives $X^{n}\in\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$. Fix $i\in[d]$. The $i$th coordinate of $X^{n}-X^{m}$ is $X^{n}_{i}-X^{m}_{i}$ by Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, and each $X^{n}_{i}$ is square-integrable by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §coordinates; that clause applied to $X^{n}-X^{m}$, which lies in $\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations, gives

the inequality by claim 6 of Properties of Finite Sums (the summands being nonnegative integrals) and the last equality by Square-Integrable Random Variables and the Mean-Square Inner Product. Both outer quantities being nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\lVert X^{n}_{i}-X^{m}_{i}\rVert_{2}\le\delta([X^{n}],[X^{m}])$. Hence, given real $\varepsilon>0$ and $N$ with $\delta([X^{n}],[X^{m}])<\varepsilon$ for all $n,m\ge N$, also $\lVert X^{n}_{i}-X^{m}_{i}\rVert_{2}<\varepsilon$ for $n,m\ge N$ by claim 2 of Elementary Order Arithmetic in an Ordered Field: the sequence $(X^{n}_{i})_{n\in\mathbb{N}}$ is Cauchy in mean square in the sense of Mean-Square Completeness of Square-Integrable Random Variables (Riesz-Fischer). That lemma, applied with the sub-$\sigma$-algebra $\mathcal{F}$ itself, provides a square-integrable random variable $Y_{i}$ with $\lVert X^{n}_{i}-Y_{i}\rVert_{2}\to0$.

Let $Y:\Omega\to\mathbb{R}^{d}$ be the map with coordinates $Y_{1},\dots,Y_{d}$, which exists by Random Vector and Its Law §coordinates; it is a random vector by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §coordinates, and lies in $\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$ by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §coordinates. As above, $s_{n}=\delta([X^{n}],[Y])^{2}=\sum_{j=1}^{d}s^{(j)}_{n}$ with $s^{(j)}_{n}=\lVert X^{n}_{j}-Y_{j}\rVert_{2}^{2}$, and each sequence $(s^{(j)}_{n})_{n}$ converges to $0$ by claim 2 of Arithmetic of Limits of Real Sequences (the square of a sequence converging to $0$). We show that $(s_{n})_{n}$ converges to $0$ by induction on the number of summands: let $T$ be the set of $k\in\mathbb{N}$ such that either $d<k$, or $k\le d$ and $(\sum_{j=1}^{k}s^{(j)}_{n})_{n}$ converges to $0$. Then $1\in T$, since $\sum_{j=1}^{1}s^{(j)}_{n}=s^{(1)}_{n}$ by claim 1 of Properties of Finite Sums; and if $k\in T$, then either $d<k$, whence $d<k+1$ by claims 5 and 1 of Properties of the Order on the Natural Numbers, or $k\le d$, and then by trichotomy (claim 3 there) either $k<d$, in which case $k+1\le d$ by claims 7 and 6 there (as $d=k+j$ for some $j$, and $1\le j$ by claim 4 there), or $k=d$; in the case $k+1\le d$, $\sum_{j=1}^{k+1}s^{(j)}_{n}=\sum_{j=1}^{k}s^{(j)}_{n}+s^{(k+1)}_{n}$ by claim 1 of Properties of Finite Sums converges to $0+0=0$ by claim 1 of Arithmetic of Limits of Real Sequences, or $k=d$, whence $d<k+1$; in every case $k+1\in T$. By Principle of Induction for the Natural Numbers, $T=\mathbb{N}$, and $d\in T$ with $d\le d$ gives $s_{n}\to0$. Finally, for real $\varepsilon>0$ one has $0<\varepsilon^{2}$ by claim 5 of Elementary Order Arithmetic in an Ordered Field, so there is $N$ with $s_{n}<\varepsilon^{2}$ for $n\ge N$, whence $\delta([X^{n}],[Y])<\varepsilon$ for $n\ge N$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; that is, $([X^{n}])_{n}$ converges to $[Y]$ in $(L^{2}(\Omega;\mathbb{R}^{d}),\delta)$, and the space is complete.

Claim 2. Let $X\in\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$ represent the class. By The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment and Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §expectation applied to the Borel map $\varphi(x)=\lVert x\rVert^{2}$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs, read as $[0,\infty]$-valued as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures,

the last equality by The Space of Square-Integrable Random Vectors §inner-product and the square root's defining property; so $\mathcal{L}(X)\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space. The law depends only on the class by The Space of Square-Integrable Random Vectors §law.

Claim 3. Let $c:\Omega\to\mathbb{R}^{d}$ be the constant map with value $a$. For $B\in\mathcal{B}(\mathbb{R}^{d})$, $c^{-1}(B)$ is $\Omega$ if $a\in B$ and $\varnothing$ otherwise, both in $\mathcal{F}$ by Sigma-Algebra and Measurable Space; so $c$ is a random vector. The random variable $\lVert c\rVert^{2}$ is the constant $\lVert a\rVert^{2}$, which equals $\lVert a\rVert^{2}\mathbf{1}_{\Omega}$, so by claim 1 of Linearity and Monotonicity of the Lebesgue Integral and The Integral of an Indicator Function is the Measure of the Set, $\mathbb{E}[\lVert c\rVert^{2}]=\lVert a\rVert^{2}P(\Omega)=\lVert a\rVert^{2}<\infty$: $c$ is square-integrable, and $\lVert c_{a}\rVert_{L^{2}}=\sqrt{\lVert a\rVert^{2}}=\lVert a\rVert$ by The Space of Square-Integrable Random Vectors §inner-product, $\lVert a\rVert$ being nonnegative by Euclidean Norm on $\mathbb{R}^n$ and the square root unique by Existence and Uniqueness of the Nonnegative Square Root. The translation $\tau_{a}$ has $k$th component $x\mapsto x_{k}+a_{k}$ by Sum of Points of $\mathbb{R}^n$, the sum of a coordinate map and a constant, Borel by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and claim 1 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets; so $\tau_{a}$ is Borel by claim 2 of that lemma. For $X\in\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$, the class $X+c_{a}$ is $[X+c]$ by The Space of Square-Integrable Random Vectors §classes, with $X+c\in\mathbf{L}^{2}(\Omega;\mathbb{R}^{d})$ by The Space of Square-Integrable Random Vectors §space, and $X+c=\tau_{a}\circ X$ pointwise; hence $\mathcal{L}(X+c_{a})=\mathcal{L}(\tau_{a}\circ X)=(\tau_{a})_{\#}\mathcal{L}(X)$ by The Space of Square-Integrable Random Vectors §law and Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §composition.