Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation

This setting fixes the standing notation for transporting Borel probability measures on a real Hilbert space with a fixed orthonormal basis when displacements are measured in the noise norm of a fixed weight sequence. It introduces no new concepts and asserts nothing beyond what the references attached to it supply and the positivity of $\bar{a}$ noted in clause 2.

1. (Background)¶ The notation and background of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation are in force: the space $X$ with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$, distance $d$, orthonormal basis $(e_{k})_{k\in\mathbb{N}}$ and coordinates $x_{k}$; the coordinate maps $\pi_{1},\pi_{2}:X\times X\to X$, and for $z\in X\times X$ we write $x=\pi_{1}(z)$ and $y=\pi_{2}(z)$; Borel sets, $\mathcal{P}(X)$, $\mathcal{P}(X\times X)$, push-forwards and weak convergence $\Rightarrow$. The second moment $M_{2}$ and the set $\mathcal{P}_{2}(X)$ of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space, the couplings $\Pi(\mu,\nu)$ and their quadratic cost $I(\pi)$, and the quadratic Wasserstein distance $W_{2}$ are those of the definitions cited, and the results Couplings on a Hilbert Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, the Lipschitz Bound and the Moment Bound, Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost, Existence of an Optimal Coupling of Two Borel Probability Measures with Finite Second Moment on a Hilbert Space, The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space and Wasserstein Convergence on a Hilbert Space: Weak Convergence, Integrals of Continuous Functions of Quadratic Growth, Convergence from Weak Convergence with Uniformly Integrable Second Moments, and Compactness are in force by reference.

2. (Noise weights)¶ $a=(a_{k})_{k\in\mathbb{N}}$ is a fixed weight sequence, called the noise weights, and $\bar{a}\in\mathbb{R}$ is a fixed number with $a_{k}\le\bar{a}$ for every $k\in\mathbb{N}$, as that definition provides; $\bar{a}$ is positive, since $0<a_{1}\le\bar{a}$.

3. (Noise space)¶ $X^{a}$ is the noise space of $a$, with the inner product $\langle\cdot,\cdot\rangle_{a}$ and norm $|\cdot|_{a}$ of The Noise Space of a Weight Sequence on a Hilbert Space with an Orthonormal Basis §inner-product; it is a real Hilbert space with the orthonormal basis $f_{k}=a_{k}^{1/2}e_{k}$ by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert and The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §basis, and The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability is in force by reference, in particular the Borel set $D_{a}\subseteq X\times X$ of pairs whose difference $y-x$ lies in $X^{a}$ and the Borel function $c_{a}$ of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §pairs, with $c_{a}(z)=|y-x|_{a}^{2}$ for $z\in D_{a}$.

4. (Square-integrable maps into the noise space)¶ For $\mu\in\mathcal{P}(X)$, $L^{2}(\mu;X^{a})$ is the space of square-integrable maps from $(X,\mathcal{B}(X),\mu)$ to $X^{a}$ of that definition, taken with $E=X^{a}$ and the basis $(f_{k})_{k\in\mathbb{N}}$, with the operations, inner product $\langle\cdot,\cdot\rangle_{\mu}$ and norm $\lVert\cdot\rVert_{\mu}$ of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations and the convention of The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §convention; it is a real Hilbert space by The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert. For $\pi\in\mathcal{P}(X\times X)$ the space $L^{2}(\pi;X^{a})$, with inner product $\langle\cdot,\cdot\rangle_{\pi}$ and norm $\lVert\cdot\rVert_{\pi}$, is read in the same way with $(X\times X,\mathcal{B}(X\times X),\pi)$. Measurability of maps into $X^{a}$ is that of The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §measurable.

5. (Reference measure)¶ $\rho\in\mathcal{P}_{2}(X)$ is a fixed Borel probability measure with finite second moment, called the reference measure.