Proof of The Constant Tuple on $L^{2}(\Omega;\mathbb{R}^{d})$: Coordinates, Projection, Tail Form, and Translation-Closed Preimages

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, $X\in L^{2}(\Omega;\mathbb{R}^{d})$, $a\in\mathbb{R}^{d}$ and $i,j\in[d]$ are arbitrary, a class and a representative of it are written alike as Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i) prescribes, and $\mathbb{E}$ is the expectation with respect to $P$. The space $L^{2}(\Omega;\mathbb{R}^{d})$ is a real Hilbert space by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (i) and $1\le d$, so once $\gamma$ is known to be orthonormal, which is Step 3, Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions applies to it as the statement describes.

Step 1 (the inner product against a constant class). By the definition of the inner product, $\langle X,c_{e_{i}}\rangle_{L^{2}}=\mathbb{E}[X\cdot c_{e_{i}}]$, where $X\cdot c_{e_{i}}$ is the random variable of Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (ii); a representative of $c_{e_{i}}$ is the constant map with value $e_{i}$ by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (iii), so the value of that random variable at $\omega\in\Omega$ is $X(\omega)\cdot e_{i}$. By Euclidean Space and Real Matrices: Standing Notation (vi) the $i$th coordinate of a point $z\in\mathbb{R}^{d}$ equals $z\cdot e_{i}$, so $X(\omega)\cdot e_{i}$ is the value at $\omega$ of the coordinate $X_{i}$; that is, $X\cdot c_{e_{i}}=X_{i}$. The law $\mathcal{L}(X)$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (ii), so The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions (i) applies to it and gives that $X_{i}$ is integrable with $\mathbb{E}[X_{i}]=m(\mathcal{L}(X))_{i}$. Hence

$$\langle X,c_{e_{i}}\rangle_{L^{2}}=m\bigl(\mathcal{L}(X)\bigr)_{i},$$

which is the first identity of clause 3.

Step 2 (clause 1). A representative of $c_{a}$ is the constant map with value $a$ (The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (iii)), so its coordinate $(c_{a})_{i}$ is the constant map $\Omega\to\mathbb{R}$ with value $a_{i}$, which is the map $a_{i}\mathbf{1}_{\Omega}$ for the indicator function $\mathbf{1}_{\Omega}$ of the set $\Omega$, an element of $\mathcal{F}$ because $\mathcal{F}$ is a $\sigma$-algebra on $\Omega$. By The Integral of an Indicator Function is the Measure of the Set the function $\mathbf{1}_{\Omega}$ is a nonnegative simple function with $\int_{\Omega}\mathbf{1}_{\Omega}\,dP=P(\Omega)$, and $P(\Omega)=1$ because $P$ is a probability measure (Probability Measures on Euclidean Space and Random Vectors: Standing Notation (vi)); being nonnegative, $\mathbf{1}_{\Omega}$ equals its own absolute value, so $\int_{\Omega}|\mathbf{1}_{\Omega}|\,dP=1$ and $\mathbf{1}_{\Omega}$ is integrable. Claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, applied with $\mathbf{1}_{\Omega}$ in both function slots and with the scalars $a_{i}$ and $0$, gives that $a_{i}\mathbf{1}_{\Omega}$ is integrable with

$$\int_{\Omega}a_{i}\mathbf{1}_{\Omega}\,dP=a_{i}\int_{\Omega}\mathbf{1}_{\Omega}\,dP+0\int_{\Omega}\mathbf{1}_{\Omega}\,dP=a_{i}.$$

Thus $\mathbb{E}[(c_{a})_{i}]=a_{i}$ by Expectation, and $m(\mathcal{L}(c_{a}))_{i}=a_{i}$ by The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions (i) applied to $c_{a}$. As $i$ was arbitrary and points of $\mathbb{R}^{d}$ with the same coordinates coincide (Euclidean Points as Tuples of Real Numbers), $m(\mathcal{L}(c_{a}))=a$. Finally $M_{2}(\mathcal{L}(c_{a}))=\lVert c_{a}\rVert_{L^{2}}^{2}=\lVert a\rVert^{2}$ by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (ii) and The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (iii). This proves clause 1.

Step 3 (clause 2). By The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (iii) and Euclidean Space and Real Matrices: Standing Notation (vi), $\lVert c_{e_{i}}\rVert_{L^{2}}=\lVert e_{i}\rVert=1$ for every $i$. Let $i\ne j$. Step 1, applied to the class $c_{e_{i}}$ and the index $j$, gives $\langle c_{e_{i}},c_{e_{j}}\rangle_{L^{2}}=m(\mathcal{L}(c_{e_{i}}))_{j}$, which equals $(e_{i})_{j}$ by clause 1; and $(e_{i})_{j}=0$, because $e_{i}$ is the point whose $i$th coordinate is $1$ and whose other coordinates are $0$ (Euclidean Space and Real Matrices: Standing Notation (vi)) and $j\ne i$. Both conditions of Orthogonality, Orthogonal Complement and Orthonormal Families in a Real Inner Product Space (iii) therefore hold for $\gamma$, which is clause 2.

Step 4 (the coordinate map). By Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions (i) the coordinate map determined by $\gamma$ sends $X$ to the point of $\mathbb{R}^{d}$ whose $i$th coordinate is $\langle X,c_{e_{i}}\rangle_{L^{2}}$ for each $i\in[d]$. By Step 1 that coordinate is $m(\mathcal{L}(X))_{i}$, so the point is $m(\mathcal{L}(X))$ by Euclidean Points as Tuples of Real Numbers. Since $m(\mathcal{L}(X))$ is determined by $\mathcal{L}(X)$, the map takes equal values at two classes with the same law.

Step 5 (the tail form as a centred second moment). Let $Z\in L^{2}(\Omega;\mathbb{R}^{d})$. By Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions (iii) one has $\lVert Z\rVert_{L^{2}}^{2}=\Pi(Z,Z)+N(Z,Z)$, where $\Pi$ is the projection form of $\gamma$ and $\Pi(Z,Z)$ is the squared Euclidean norm of the image of $Z$ under the coordinate map, hence $\lVert m(\mathcal{L}(Z))\rVert^{2}$ by Step 4; and $\lVert Z\rVert_{L^{2}}^{2}=M_{2}(\mathcal{L}(Z))$ by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (ii). Therefore

$$N(Z,Z)=M_{2}\bigl(\mathcal{L}(Z)\bigr)-\bigl\lVert m(\mathcal{L}(Z))\bigr\rVert^{2}.$$

Step 6 (the identity $\Lambda^{\sharp}a=c_{a}$). Apply Step 5 to $c_{a}$: by clause 1 both $M_{2}(\mathcal{L}(c_{a}))$ and $\lVert m(\mathcal{L}(c_{a}))\rVert^{2}$ equal $\lVert a\rVert^{2}$, so $N(c_{a},c_{a})=0$. By Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions (iii) also $N(c_{a},c_{a})=\lVert c_{a}-Jc_{a}\rVert_{L^{2}}^{2}$, so the nonnegative real number $\lVert c_{a}-Jc_{a}\rVert_{L^{2}}$ has square $0=0\cdot0$ and is therefore $0$ by claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; hence $c_{a}-Jc_{a}$ is the zero vector by Elementary Identities in a Real Inner Product Space (iii), and $c_{a}=Jc_{a}$, $L^{2}(\Omega;\mathbb{R}^{d})$ being a real vector space. On the other hand $J$ is the coordinate map followed by $\Lambda^{\sharp}$ (Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions (i)), so $Jc_{a}=\Lambda^{\sharp}\bigl(m(\mathcal{L}(c_{a}))\bigr)=\Lambda^{\sharp}a$ by Step 4 and clause 1. Therefore $\Lambda^{\sharp}a=c_{a}$.

Step 7 (the remaining assertions of clauses 3 and 4). By Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions (i) the dot product of $a$ with the image of $X$ under the coordinate map equals $\langle\Lambda^{\sharp}a,X\rangle_{L^{2}}$; by Steps 4 and 6 this reads $a\cdot m(\mathcal{L}(X))=\langle c_{a},X\rangle_{L^{2}}$, the second identity of clause 3. Together with Steps 4 and 6 this proves clause 3.

For clause 4, $JX=\Lambda^{\sharp}\bigl(m(\mathcal{L}(X))\bigr)=c_{m(\mathcal{L}(X))}$ by the description of $J$ in Step 6 and by Steps 4 and 6. Hence, by Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions (iii),

$$N(X,X)=\lVert X-JX\rVert_{L^{2}}^{2}=\bigl\lVert X-c_{m(\mathcal{L}(X))}\bigr\rVert_{L^{2}}^{2}.$$

The law of $X-c_{m(\mathcal{L}(X))}$ is $\overline{\mathcal{L}(X)}$ by The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions (ii), so the right-hand side equals $M_{2}(\overline{\mathcal{L}(X)})$ by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (ii); and it equals $M_{2}(\mathcal{L}(X))-\lVert m(\mathcal{L}(X))\rVert^{2}$ by Step 5. Each of these expressions is determined by $\mathcal{L}(X)$, so $N(X,X)$ takes equal values at two classes with the same law. This proves clause 4.

Step 8 (clause 5). Assume that $(\Omega,\mathcal{F},P)$ is rich and let $Q$ be as in clause 5. Since $Q$ is nonempty there is $\mu\in Q$, and by The Wasserstein Distance and the Mean-Square Distance of Random Vectors (iii) there is $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\mu$; then $X\in Q^{\Lambda}$, so $Q^{\Lambda}$ is nonempty. Let $X\in Q^{\Lambda}$ and $a\in\mathbb{R}^{d}$. By Step 6, $X+\Lambda^{\sharp}a=X+c_{a}$, and $\mathcal{L}(X+c_{a})=(\tau_{a})_{\#}\mathcal{L}(X)$ by The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (iii), which lies in $Q$ by the hypothesis on $Q$, since $\mathcal{L}(X)\in Q$. Hence $X+\Lambda^{\sharp}a\in Q^{\Lambda}$, and $Q^{\Lambda}$ is translation-closed along $\gamma$ by Subsets of a Real Hilbert Space Translation-Closed along an Orthonormal Tuple (i).

Step 9 (clause 6). The penalty domain $\mathcal{D}$ contains the score domain $\mathcal{D}_{\Sigma}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i), and $\mathcal{D}_{\Sigma}$ is nonempty by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (ii), so $\mathcal{D}$ is nonempty; and $(\tau_{a})_{\#}\mu\in\mathcal{D}$ for every $\mu\in\mathcal{D}$ and every $a\in\mathbb{R}^{d}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (iv). Clause 5, applied with $\mathcal{D}$ in the role of $Q$, gives that $\mathcal{D}^{\Lambda}$ is translation-closed along $\gamma$. This proves clause 6 and completes the proof of the lemma.