Penalty Pairs with Closed Score

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$. For $\nu\in\mathcal{D}_{\Sigma}$ the score $\Sigma(\nu)$ lies in $T_{\nu}$, hence in the space $L^{2}(\nu;\mathbb{R}^{d})$ with its norm $\lVert\cdot\rVert_{\nu}$; for $X\in L^{2}(\Omega;\mathbb{R}^{d})$ with $\mathcal{L}(X)=\nu$, the composition $\Sigma(\nu)\circ X$ is the element of $L^{2}(\Omega;\mathbb{R}^{d})$ of that clause, which satisfies $\lVert\Sigma(\nu)\circ X\rVert_{L^{2}}=\lVert\Sigma(\nu)\rVert_{\nu}$. Convergence of a sequence in $L^{2}(\Omega;\mathbb{R}^{d})$ is that of the metric $d_{L^{2}}$ fixed in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (i), and weak convergence of a sequence in the real inner product space $L^{2}(\Omega;\mathbb{R}^{d})$ is that of that definition.

(Closed score)(i) The penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ has closed score if the following holds for every nonnegative $R\in\mathbb{R}$. Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence in $L^{2}(\Omega;\mathbb{R}^{d})$ such that $\mathcal{L}(X_{n})\in\mathcal{D}_{\Sigma}$ and

$$\bigl\lVert\Sigma(\mathcal{L}(X_{n}))\bigr\rVert_{\mathcal{L}(X_{n})}\le R$$

for every $n\in\mathbb{N}$, and suppose that $(X_{n})_{n\in\mathbb{N}}$ converges to $X$ in $L^{2}(\Omega;\mathbb{R}^{d})$. Then $\mathcal{L}(X)\in\mathcal{D}_{\Sigma}$, and the sequence whose $n$-th term is $\Sigma(\mathcal{L}(X_{n}))\circ X_{n}$ converges weakly to $\Sigma(\mathcal{L}(X))\circ X$.