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

Each result cited is universally quantified over the data in its own statement. A random variable is a map $\Omega\to\mathbb{R}$ measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R})$ by Probability Spaces, Events and Random Elements. Each Euclidean space $\mathbb{R}^{m}$ is a metric space under $d_{E}$ whose Borel $\sigma$-algebra in the sense of Borel Sigma-Algebra of a Metric Space is $\mathcal{B}(\mathbb{R}^{m})$ by The Borel $\sigma$-Algebras of Euclidean Space and of the Euclidean Metric Coincide, so that claim 4 of Borel Measurability and Bounded Integration on a Metric Space, cited below as the composition rule, applies to compositions of a map $\Omega\to\mathbb{R}^{m}$ measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R}^{m})$ with a Borel map on $\mathbb{R}^{m}$. The proofs of claims 1 to 4 do not use claim 5; in the proof of claim 5, claims 2 and 3 are applied to the lemma instantiated at the dimension $d+d$ in place of $d$.

Claim 1. By Probability Measures on Euclidean Space and Random Vectors: Standing Notation (i), $\mathcal{B}(\mathbb{R}^{d})$ is the $\sigma$-algebra $\mathcal{B}_{d}$ of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and claim 2 of that lemma, applied to the measurable space $(\Omega,\mathcal{F})$ and $f=Z$, states that $Z$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}_{d}$ if and only if each coordinate $Z_{i}$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R})$, that is, is a random variable.

Claim 2. If $g:\mathbb{R}^{d}\to\mathbb{R}^{m}$ is Borel, then $g\circ X$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R}^{m})$ by the composition rule, hence a random vector in $\mathbb{R}^{m}$, and for $B\in\mathcal{B}(\mathbb{R}^{m})$,

$$\mathcal{L}(g\circ X)(B)=P\bigl((g\circ X)^{-1}(B)\bigr)=P\bigl(X^{-1}(g^{-1}(B))\bigr)=\mathcal{L}(X)\bigl(g^{-1}(B)\bigr)=g_{\#}\mathcal{L}(X)(B),$$

by Random Vector and Its Law (iii) and Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), $g^{-1}(B)\in\mathcal{B}(\mathbb{R}^{d})$ because $g$ is Borel. For Borel $\varphi:\mathbb{R}^{d}\to\mathbb{R}$, $\varphi\circ X$ is a random variable by the composition rule. For Borel $\varphi:\mathbb{R}^{d}\to[0,\infty]$ and real $c$, $\{\omega:\varphi(X(\omega))>c\}=X^{-1}(\{x:\varphi(x)>c\})$, the preimage under $X$ of a member of $\mathcal{B}(\mathbb{R}^{d})$, lies in $\mathcal{F}$; so $\varphi\circ X$ is measurable in the sense of Measurable Functions and the Lebesgue Integral: Standing Notation (iii). The pairing $(X,Y):\Omega\to\mathbb{R}^{d+d}$ is measurable with respect to $\mathcal{F}$ and $\mathcal{B}(\mathbb{R}^{d+d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v). The map $A:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$, $A(z)=\mathrm{pr}_{1}(z)+\mathrm{pr}_{2}(z)$, has $k$th component $z\mapsto z_{k}+z_{d+k}$ by Sum of Points of $\mathbb{R}^n$, because $z=\iota(\mathrm{pr}_{1}(z),\mathrm{pr}_{2}(z))$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) and the $k$th components of $\mathrm{pr}_{1}(z)$ and $\mathrm{pr}_{2}(z)$ are $z_{k}$ and $z_{d+k}$ by the description of $\iota$ recorded in the preamble of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets; this is a sum of two coordinate maps, Borel by 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 and claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps; so $A$ is Borel by claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and $X+Y=A\circ(X,Y)$ is a random vector by the composition rule, since $\mathrm{pr}_{1}((X,Y)(\omega))=X(\omega)$ and $\mathrm{pr}_{2}((X,Y)(\omega))=Y(\omega)$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i). The same argument with $z\mapsto\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)$, whose $k$th component is $z_{k}-z_{d+k}$ by Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, shows that $X-Y$ is a random vector, and $aX$ is the composition of $X$ with the map $x\mapsto ax$, whose $k$th component $x\mapsto ax_{k}$ (Scalar Multiple of a Point of $\mathbb{R}^n$) is a scalar multiple of a coordinate map, Borel by 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 and claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, so that $x\mapsto ax$ is Borel by claim 2 of the former. The maps $\lVert X\rVert$ and $\lVert X\rVert^{2}$ are the compositions of $X$ with the Borel maps $x\mapsto\lVert x\rVert$ and $x\mapsto\lVert x\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv), hence random variables; and $X\cdot Y$ and $\lVert X-Y\rVert^{2}$ are random variables by the consequence for measurable maps recorded in that clause, applied with $(E,\mathcal{E})=(\Omega,\mathcal{F})$, $u=X$ and $v=Y$.

Claim 3. Both assertions are claim 2 of Image Measures and Measures with a Density: Measures, Integrals and Change of Variables, applied to the measure space $(\Omega,\mathcal{F},P)$, the measurable map $T=X$ into $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$ and $g=\varphi$: the image measure there is $\mathcal{L}(X)$ by Random Vector and Its Law (iii), and the integral $\int_{\Omega}\varphi\circ X\,dP$ is $\mathbb{E}[\varphi\circ X]$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (vi), in the $[0,\infty]$-valued reading for the first assertion and as the expectation of the integrable random variable $\varphi\circ X$ for the second.

Claim 4. Let $D=\lVert X-Y\rVert^{2}$, a random variable by claim 2 with nonnegative values by claim 2 of Nonnegativity of Squares in an Ordered Field. For $\omega\in\Omega$, $X(\omega)=Y(\omega)$ holds exactly when $X(\omega)-Y(\omega)=0_{\mathbb{R}^{d}}$ (claim 2 of Elementary Identities in a Vector Space), exactly when $\lVert X(\omega)-Y(\omega)\rVert=0$ (claim 3 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square), exactly when $D(\omega)=0$ (claims 1 and 3 of Zero Products and Elementary Identities in a Field), and, $D(\omega)$ being nonnegative, exactly when $D(\omega)>0$ fails. Hence $\{X=Y\}=\Omega\setminus\{D>0\}$, and $\{D>0\}\in\mathcal{F}$ by Measurable Functions and the Lebesgue Integral: Standing Notation (iii), so $\{X=Y\}\in\mathcal{F}$ by σ-Algebra and Measurable Space. Now suppose $P(X=Y)=1$ and let $B\in\mathcal{B}(\mathbb{R}^{d})$. Write $E=\{X=Y\}$. By claim 3 of Basic Properties of a Measure, $P(\Omega\setminus E)=1-1=0$, and by claim 2 there $P(\{X\in B\}\setminus E)\le P(\Omega\setminus E)=0$ and likewise $P(\{Y\in B\}\setminus E)=0$. The sets $\{X\in B\}\cap E$ and $\{X\in B\}\setminus E$ are disjoint with union $\{X\in B\}$, so $P(X\in B)=P(\{X\in B\}\cap E)$ by claim 1 of Basic Properties of a Measure; likewise $P(Y\in B)=P(\{Y\in B\}\cap E)$. On $E$ the values of $X$ and $Y$ agree, so $\{X\in B\}\cap E=\{Y\in B\}\cap E$. Therefore $\mathcal{L}(X)(B)=P(X\in B)=P(Y\in B)=\mathcal{L}(Y)(B)$.

Claim 5. $(X,Y)$ is a random vector in $\mathbb{R}^{d+d}$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v), as noted in the proof of claim 2. By claim 2 applied with $g=\mathrm{pr}_{1}$, which is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v), $(\mathrm{pr}_{1})_{\#}\mathcal{L}((X,Y))=\mathcal{L}(\mathrm{pr}_{1}\circ(X,Y))=\mathcal{L}(X)$, since $\mathrm{pr}_{1}\circ(X,Y)=X$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i); likewise $(\mathrm{pr}_{2})_{\#}\mathcal{L}((X,Y))=\mathcal{L}(Y)$. So $\mathcal{L}((X,Y))\in\Pi(\mathcal{L}(X),\mathcal{L}(Y))$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (i). Let $\varphi:\mathbb{R}^{d+d}\to[0,\infty]$ be the map $z\mapsto\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert^{2}$, Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v). By Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost (ii) and claim 3 applied to the random vector $(X,Y)$,

$$I\bigl(\mathcal{L}((X,Y))\bigr)=\int_{\mathbb{R}^{d+d}}\varphi\,d\mathcal{L}((X,Y))=\mathbb{E}[\varphi\circ(X,Y)]=\mathbb{E}[\lVert X-Y\rVert^{2}],$$

because $\varphi((X,Y)(\omega))=\lVert X(\omega)-Y(\omega)\rVert^{2}$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i). Conversely, let $Z$ be a random vector in $\mathbb{R}^{d+d}$. Then $\mathrm{pr}_{1}\circ Z$ and $\mathrm{pr}_{2}\circ Z$ are random vectors in $\mathbb{R}^{d}$ by the composition rule, and for every $\omega$, $Z(\omega)=\iota(\mathrm{pr}_{1}(Z(\omega)),\mathrm{pr}_{2}(Z(\omega)))=(\mathrm{pr}_{1}\circ Z,\mathrm{pr}_{2}\circ Z)(\omega)$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) and the definition of the pairing in Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (ii).