The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation

This setting fixes the standing notation used by results on real-valued functions on the quadratic Wasserstein space and on their lifts to the space of square-integrable random vectors. It is layered on Probability Measures on Euclidean Space and Random Vectors: Standing Notation and on Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus, hence on Real Hilbert Spaces: Standing Notation and Background, whose notation is in force throughout; it introduces no new concept and asserts nothing beyond the identifications recorded below, each of which is justified by the reference attached to it.

1. (Dimension and probability space)(i) $d\in\mathbb{N}$ with $1\le d$ is fixed, and $(\Omega,\mathcal{F},P)$ is the probability space fixed there. Random vectors on $(\Omega,\mathcal{F},P)$, their laws $\mathcal{L}(X)$ and the shorthand $X\sim\mu$ are as defined there, and Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality and Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form are in force. The letter $d$ is reserved for the dimension: the distance of every metric space named below carries its ambient subscript ($d_{L^{2}}$ below, $d_{\mathbb{R}}$ on $\mathbb{R}$ as in Real Hilbert Spaces: Standing Notation and Background (i), and the Euclidean distance $d_{E}$ and Euclidean norm $\lVert\cdot\rVert$ on $\mathbb{R}^{d}$ as in Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background (ii)), and the symbol $\mathcal{L}$ applied to a random vector or to a class always denotes its law, never the space of bounded linear maps of Real Hilbert Spaces: Standing Notation and Background (iv).

2. (Richness)(ii) That $(\Omega,\mathcal{F},P)$ is rich is the property of Rich Probability Space (i); this setting does not assume it, and a result that needs it says so.

3. (The space of square-integrable random vectors)(iii) $L^{2}(\Omega;\mathbb{R}^{d})$ is the space of classes of square-integrable random vectors in $\mathbb{R}^{d}$ on $(\Omega,\mathcal{F},P)$, with the inner product and norm $\langle\cdot,\cdot\rangle_{L^{2}}$, $\lVert\cdot\rVert_{L^{2}}$, the law $\mathcal{L}(X)$ of a class and the notational convention fixed there, under which a class and a representative of it are denoted by the same symbol. It 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 it is one of the real Hilbert spaces named by this setting, hence by every result adopting it, so that the notation of that setting and the differential calculus on open subsets of it are in force for it, with the following reading: the norm written $|\cdot|_{L^{2}}$ there is written $\lVert\cdot\rVert_{L^{2}}$ here and the inner product is $\langle\cdot,\cdot\rangle_{L^{2}}$, and the distance is written $d_{L^{2}}(X,Y)=\lVert X-Y\rVert_{L^{2}}$. The set $L^{2}(\Omega;\mathbb{R}^{d})$ is an open subset of itself by Real Hilbert Spaces: Standing Notation and Background (iii).

4. (The Wasserstein space)(iv) $\mathcal{P}_{2}(\mathbb{R}^{d})$ is the set of probability measures on $\mathbb{R}^{d}$ with finite second moment, $W_{2}$ is the quadratic Wasserstein distance, a metric on $\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space (iv), and $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ is the quadratic Wasserstein space named there. Couplings, the set $\Pi(\mu,\nu)$ and the quadratic cost $I(\pi)$ are as defined there, the symbol $I$ applied to a coupling always denoting its cost and never the identity form of Real Hilbert Spaces: Standing Notation and Background (v); Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound is in force. For a function $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, that $u$ is Lipschitz with a constant, uniformly continuous or continuous is understood with the metric $W_{2}$ on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and the metric $d_{\mathbb{R}}$ on $\mathbb{R}$; for a function on $L^{2}(\Omega;\mathbb{R}^{d})$ the same three words are understood with the metric $d_{L^{2}}$, as fixed in Real Hilbert Spaces: Standing Notation and Background (iii). A real-valued function on either space is bounded as defined there.

5. (The law map)(v) $\Lambda:L^{2}(\Omega;\mathbb{R}^{d})\to\mathcal{P}_{2}(\mathbb{R}^{d})$, $\Lambda(X)=\mathcal{L}(X)$, is the law map of that lemma, which is in force; the symbol $\Lambda$ always denotes the law map, and a Lipschitz constant, written $\Lambda$ in Lipschitz Map Between Metric Spaces, is written with another letter here. In particular $W_{2}(\mathcal{L}(X),\mathcal{L}(Y))\le\lVert X-Y\rVert_{L^{2}}$ for all $X,Y\in L^{2}(\Omega;\mathbb{R}^{d})$ by The Wasserstein Distance and the Mean-Square Distance of Random Vectors (ii), and if $(\Omega,\mathcal{F},P)$ is rich, then $\Lambda$ is surjective by The Wasserstein Distance and the Mean-Square Distance of Random Vectors (iii) and $W_{2}(\mu,\nu)$ is the infimum of $\lVert X-Y\rVert_{L^{2}}$ over $X\sim\mu$, $Y\sim\nu$ by The Wasserstein Distance and the Mean-Square Distance of Random Vectors (iv).

6. (Constants and translations)(vi) For $a\in\mathbb{R}^{d}$, $c_{a}\in L^{2}(\Omega;\mathbb{R}^{d})$ is the class of the constant map with value $a$, and $\tau_{a}:\mathbb{R}^{d}\to\mathbb{R}^{d}$, $\tau_{a}(x)=x+a$, is the translation by $a$, as in The Space of Square-Integrable Random Vectors is a Real Hilbert Space; Its Laws Have Finite Second Moment; Constants and Translations (iii); by that clause $\lVert c_{a}\rVert_{L^{2}}=\lVert a\rVert$ and $\mathcal{L}(X+c_{a})=(\tau_{a})_{\#}\mathcal{L}(X)$ for every $X\in L^{2}(\Omega;\mathbb{R}^{d})$.