Proof of The Joint Law of Two Classes of Square-Integrable Random Vectors: Marginals, Cost, and Invariance Under Shifts by a Vector Field

Each result cited is universally quantified over the data in its own statement and is applied here to the data named in the statement of the lemma. Throughout, a symbol for a class also denotes a chosen representative, as the convention allows, and it is said explicitly when a statement is about representatives. Push-forwards of measures by Borel maps between Euclidean spaces are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward. For a random vector $W$ on $(\Omega,\mathcal{F},P)$ three facts are used: $T\circ W$ is a random vector with $\mathcal{L}(T\circ W)=T_{\#}\mathcal{L}(W)$ for Borel $T$, by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §composition; $\int\varphi\,d\mathcal{L}(W)=\int_{\Omega}\varphi\circ W\,dP$ for nonnegative Borel $\varphi$, by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §expectation; and two random vectors agreeing almost surely have the same law, by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §almost-sure. The coordinate projections $\mathrm{pr}_{1},\mathrm{pr}_{2}$ are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, the pairing of two Borel maps is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, and compositions of Borel maps are Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, $\mathrm{pr}_{1}\circ(X,V)=X$ and $\mathrm{pr}_{2}\circ(X,V)=V$ for any two representatives.

Claim 1. Let $X,V$ be representatives; their pairing $(X,V)$ is a random vector in $\mathbb{R}^{d+d}$ by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §pair.

Independence of the representatives. Let $X_{1},V_{1}$ be other representatives of the same classes, so that $P(X=X_{1})=1$ and $P(V=V_{1})=1$ by The Space of Square-Integrable Random Vectors §classes. The complement of $\{X=X_{1}\}\cap\{V=V_{1}\}$ is the union of the two complements, each of measure $0$ by claim 3 of Basic Properties of a Measure; by claim 4 of that lemma, applied to the sequence whose first two terms are these complements and whose remaining terms are the empty set, the union has measure at most the sum of the measures of the terms, which is $0$ because the first two terms have measure $0$ and $P(\varnothing)=0$; so the union has measure $0$, and the intersection has measure $1$ by claim 3 again. At every point of that intersection the random vectors $(X,V)$ and $(X_{1},V_{1})$ take the same value, so the event $\{(X,V)=(X_{1},V_{1})\}$ of Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §almost-sure contains the intersection and has probability at least $1$ by claim 2 of Basic Properties of a Measure, hence probability $1$ since $P(\Omega)=1$; hence $\mathcal{L}((X,V))=\mathcal{L}((X_{1},V_{1}))$ by Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §almost-sure. This defines $\mathcal{L}(X,V)$.

Second moment. The pairing $(X,V)$ has value $\iota(X(\omega),V(\omega))$ at $\omega$, with $\iota$ the concatenation map of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, and by claim 3 of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space, $\lVert\iota(y,w)\rVert^{2}=\lVert y\rVert^{2}+\lVert w\rVert^{2}$ for $y,w\in\mathbb{R}^{d}$; so the nonnegative Borel function $\lVert\cdot\rVert^{2}$ on $\mathbb{R}^{d+d}$ (Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions) composed with $(X,V)$ has value $\lVert X\rVert^{2}+\lVert V\rVert^{2}$ at each point of $\Omega$. By Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §expectation, the additivity of the integral of nonnegative functions (claim 1 of Linearity and Monotonicity of the Lebesgue Integral) and The Space of Square-Integrable Random Vectors §inner-product,

the second moment being the integral of $\lVert\cdot\rVert^{2}$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment. The right-hand side is a real number, so $\mathcal{L}(X,V)\in\mathcal{P}_{2}(\mathbb{R}^{d+d})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §space. This completes claim 1.

Claim 2. By Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §pair, applied to representatives, the law $\mathcal{L}(X,V)$ of the pairing is a coupling of $\mathcal{L}(X)$ and $\mathcal{L}(V)$ with quadratic cost $I(\mathcal{L}(X,V))=\mathbb{E}[\lVert X-V\rVert^{2}]$; here the laws of the representatives are the laws of the classes by The Space of Square-Integrable Random Vectors §law. By Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling a coupling of $\mathcal{L}(X)$ and $\mathcal{L}(V)$ is a probability measure on $\mathbb{R}^{d+d}$ whose push-forwards by $\mathrm{pr}_{1}$ and by $\mathrm{pr}_{2}$ are $\mathcal{L}(X)$ and $\mathcal{L}(V)$, which gives the two displayed marginal identities. Finally the representative $X-V$ of the class $X-V$ (The Space of Square-Integrable Random Vectors §classes) satisfies $\mathbb{E}[\lVert X-V\rVert^{2}]=\lVert X-V\rVert_{L^{2}}^{2}$ by The Space of Square-Integrable Random Vectors §inner-product. This completes claim 2.

Claim 3. Suppose $\mathcal{L}(X,V)=\mathcal{L}(X',V')$. By claim 2, $\mathcal{L}(X)=(\mathrm{pr}_{1})_{\#}\mathcal{L}(X,V)=(\mathrm{pr}_{1})_{\#}\mathcal{L}(X',V')=\mathcal{L}(X')$. Write $\mu$ for this common law; then $L^{2}(\mathcal{L}(X);\mathbb{R}^{d})$ and $L^{2}(\mathcal{L}(X');\mathbb{R}^{d})$ are the same space $L^{2}(\mu;\mathbb{R}^{d})$, so $\eta\circ X'$ is defined by Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity §composition. Let $\eta\in L^{2}(\mu;\mathbb{R}^{d})$ and $t\in\mathbb{R}$, and let $\sigma$ be a representative of $\eta$, a Borel map $\mathbb{R}^{d}\to\mathbb{R}^{d}$ by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields. By Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity §composition the class $\eta\circ X$ is the class of the random vector $\sigma\circ X$ formed from representatives, and likewise $\eta\circ X'$ is the class of $\sigma\circ X'$; so by The Space of Square-Integrable Random Vectors §classes the classes $V+t\,\eta\circ X$ and $V'+t\,\eta\circ X'$ have the representatives $V+t\,(\sigma\circ X)$ and $V'+t\,(\sigma\circ X')$, formed pointwise from representatives $X,V,X',V'$.

Let $S:\mathbb{R}^{d+d}\to\mathbb{R}^{d+d}$ be the pairing of $\mathrm{pr}_{1}$ with the pointwise sum of $\mathrm{pr}_{2}$ and $t\,(\sigma\circ\mathrm{pr}_{1})$, that is, $S(z)=\bigl(\mathrm{pr}_{1}(z),\ \mathrm{pr}_{2}(z)+t\,\sigma(\mathrm{pr}_{1}(z))\bigr)$. The map $\sigma\circ\mathrm{pr}_{1}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ is Borel as a composition of Borel maps. The map $H=\mathrm{pr}_{2}+t\,(\sigma\circ\mathrm{pr}_{1}):\mathbb{R}^{d+d}\to\mathbb{R}^{d}$ is Borel by the componentwise criterion, 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 (whose $\sigma$-algebras are the Borel $\sigma$-algebras by claim 5 of that lemma), writing $f^{i}$ for the $i$th component map of a map $f$ into $\mathbb{R}^{d}$ as that claim does: the $i$th component of $H$ is $\mathrm{pr}_{2}^{i}+t\,(\sigma\circ\mathrm{pr}_{1})^{i}$, the components $\mathrm{pr}_{2}^{i}$ and $(\sigma\circ\mathrm{pr}_{1})^{i}$ are Borel real-valued functions by the same claim 2 applied to the Borel maps $\mathrm{pr}_{2}$ and $\sigma\circ\mathrm{pr}_{1}$, and a sum of a Borel real-valued function with a real multiple of another is Borel by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Finally $S=(\mathrm{pr}_{1},H)$ is the pairing of two Borel maps, Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing. At every $\omega\in\Omega$, using $\mathrm{pr}_{1}\circ(X,V)=X$ and $\mathrm{pr}_{2}\circ(X,V)=V$,

so $S\circ(X,V)=(X,V+t\,(\sigma\circ X))$ as maps on $\Omega$, and in the same way $S\circ(X',V')=(X',V'+t\,(\sigma\circ X'))$. By Basic Properties of Random Vectors: Coordinates, Borel Images and Arithmetic, Change of Variables, Almost Sure Equality and Pairs §composition and claim 1, computed with these representatives,

the middle equality being the hypothesis. This completes claim 3.