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

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, the space $L^{2}(\Omega;\mathbb{R}^{d})$ with its inner product $\langle\cdot,\cdot\rangle_{L^{2}}$, norm $\lVert\cdot\rVert_{L^{2}}$, laws $\mathcal{L}(X)$, constant classes $c_{a}$ and translations $\tau_{a}$, the second moment $M_{2}$, the Wasserstein space $\mathcal{P}_{2}(\mathbb{R}^{d})$, the standard basis vectors $e_{1},\dots,e_{d}$ of $\mathbb{R}^{d}$ and the preimage $Q^{\Lambda}$ of a subset $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ under the law map $\Lambda$ are as fixed there; push-forwards are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv). The mean $m(\mu)\in\mathbb{R}^{d}$ and the centred measure $\bar{\mu}$ of $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ are those of that lemma.

Let $\gamma=(c_{e_{1}},\dots,c_{e_{d}})$, a $d$-tuple in $L^{2}(\Omega;\mathbb{R}^{d})$. By clause 2 below $\gamma$ is orthonormal, so Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions applies to it, read with $L^{2}(\Omega;\mathbb{R}^{d})$ in place of $H$ and with $d$ in place of the dimension written $m$ there: the coordinate map and the map $\Lambda^{\sharp}$ and the tail form $N$ determined by $\gamma$ are those of that lemma. Two of its symbols are renamed here, because this setting binds them otherwise: its projection, there written $P$, is written $J$, the letter $P$ being the probability measure; and its coordinate map, there written $\Lambda$, is referred to in words, the letter $\Lambda$ being the law map. Being translation-closed along $\gamma$ is as defined there. Then the following hold.

1. (Constant classes)(i) For every $a\in\mathbb{R}^{d}$,

$$m\bigl(\mathcal{L}(c_{a})\bigr)=a,\qquad M_{2}\bigl(\mathcal{L}(c_{a})\bigr)=\lVert a\rVert^{2}.$$

2. (The tuple is orthonormal)(ii) The $d$-tuple $\gamma$ is orthonormal.

3. (Coordinates are the mean of the law)(iii) For every $X\in L^{2}(\Omega;\mathbb{R}^{d})$, every $i\in[d]$ and every $a\in\mathbb{R}^{d}$,

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

the coordinate map determined by $\gamma$ sends $X$ to $m(\mathcal{L}(X))$, and $\Lambda^{\sharp}a=c_{a}$. In particular that map takes equal values at two classes with the same law.

4. (The projection and the tail form)(iv) For every $X\in L^{2}(\Omega;\mathbb{R}^{d})$,

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

In particular $N(X,X)$ takes equal values at two classes with the same law.

5. (Preimages of translation-invariant sets)(v) Assume that $(\Omega,\mathcal{F},P)$ is rich, and let $Q\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ be nonempty and satisfy $(\tau_{a})_{\#}\mu\in Q$ for every $\mu\in Q$ and every $a\in\mathbb{R}^{d}$. Then $Q^{\Lambda}$ is translation-closed along $\gamma$.

6. (Penalty domains)(vi) Assume that $(\Omega,\mathcal{F},P)$ is rich and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$. Then $\mathcal{D}^{\Lambda}$ is translation-closed along $\gamma$.