Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost

In the setting of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation, couplings, the sets $\Pi(\mu,\nu)$ and the quadratic cost $I$ are those of that definition. Then the following hold.

1. (Tightness)¶ Let $\mathcal{M},\mathcal{N}\subseteq\mathcal{P}(X)$ be tight in $(X,d)$. Then the set of all $\pi\in\Pi(\mu,\nu)$ with $\mu\in\mathcal{M}$ and $\nu\in\mathcal{N}$ is tight in $X\times X$. In particular, since $\{\mu\}$ and $\{\nu\}$ are tight by Ulam's Theorem: a Finite Borel Measure on a Complete Separable Metric Space is Tight §tight, for all $\mu,\nu\in\mathcal{P}(X)$ the set $\Pi(\mu,\nu)$ is tight in $X\times X$.

2. (Closedness)¶ Let $\mu_{j},\nu_{j},\mu,\nu\in\mathcal{P}(X)$ for $j\in\mathbb{N}$ with $\mu_{j}\Rightarrow\mu$ and $\nu_{j}\Rightarrow\nu$, let $\pi_{j}\in\Pi(\mu_{j},\nu_{j})$, and let $\pi\in\mathcal{P}(X\times X)$ with $\pi_{j}\Rightarrow\pi$. Then $\pi\in\Pi(\mu,\nu)$.

3. (Lower semicontinuity of the cost)¶ In the situation of claim 2, so that $\pi\in\Pi(\mu,\nu)$ and $I(\pi)$ is defined, suppose that every $I(\pi_{j})$ is a real number and that the sequence $(I(\pi_{j}))_{j\in\mathbb{N}}$ is bounded. Then $I(\pi)<\infty$ and $I(\pi)\le\liminf_{j}I(\pi_{j})$, the limit inferior of that bounded sequence.