Proof of The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity

Each result cited is universally quantified over the data in its own statement.

Throughout, the notation is that of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation, whose clause Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §background puts the notation of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation in force. Since the letters $\pi_{1},\pi_{2}$ denote the coordinate maps of $X\times X$, couplings in sequences are written $\gamma_{j}$; $\sigma:X\times X\to X\times X$ is the swap $\sigma(x,y)=(y,x)$ of Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §swap, and gluings in the sense of Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §glued are written $\theta$, with the maps $q_{1},q_{2},q_{3}$ of Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs. Square roots are the nonnegative square roots of Existence and Uniqueness of the Nonnegative Square Root. That $s\mapsto\sqrt{s}$ is increasing on $[0,\infty)$ follows from claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field; for $s,t\ge0$ one has $\sqrt{s^{2}}=s$, and $\sqrt{st}=\sqrt{s}\sqrt{t}$ because $\sqrt{s}\sqrt{t}\ge0$ and $(\sqrt{s}\sqrt{t})^{2}=st$, both by the uniqueness in that theorem; and the Archimedean property is The Archimedean Property of the Real Numbers. These three results are in force by The Real Numbers: Standing Notation and Background §background. For noise-connected $\mu,\nu\in\mathcal{P}(X)$ we use repeatedly that, by The Noise Wasserstein Distance §distance, $W_{a}(\mu,\nu)^{2}$ is the greatest lower bound of the nonempty set $\{I^{a}(\gamma):\gamma\in\Pi^{a}(\mu,\nu)\}$ of nonnegative reals and $W_{a}(\mu,\nu)^{2}\le I^{a}(\gamma)$ for every $\gamma\in\Pi^{a}(\mu,\nu)$.

Preliminaries. We record four elementary facts used below.

Claim (subsequences). Let $Y$ be $X$ or $X\times X$, let $\kappa_{n},\kappa\in\mathcal{P}(Y)$ for $n\in\mathbb{N}$ with $\kappa_{n}\Rightarrow\kappa$, and let $(n_{k})_{k\in\mathbb{N}}$ be strictly increasing in $\mathbb{N}$. Then $\kappa_{n_{k}}\Rightarrow\kappa$. Indeed, fix a bounded continuous $f:Y\to\mathbb{R}$. By Weak Convergence of Finite Borel Measures on a Metric Space the real sequence $(\int_{Y}f\,d\kappa_{n})_{n}$ converges to $\int_{Y}f\,d\kappa$ in the sense of Limit of a Sequence of Real Numbers; this is convergence in the real line of The Absolute Value Metric on the Real Line in the sense of Convergent Sequence in a Metric Space, the two definitions requiring the same inequality $|\int_{Y}f\,d\kappa_{n}-\int_{Y}f\,d\kappa|<\varepsilon$ for all large $n$. By A Subsequence of a Convergent Sequence Has the Same Limit the subsequence $(\int_{Y}f\,d\kappa_{n_{k}})_{k}$ converges to $\int_{Y}f\,d\kappa$, and as $f$ was arbitrary, $\kappa_{n_{k}}\Rightarrow\kappa$.

Claim (constant sequences). For $\kappa\in\mathcal{P}(Y)$ the constant sequence with all terms $\kappa$ converges weakly to $\kappa$: for every bounded continuous $f$ the sequence of integrals is constant, equal to $\int_{Y}f\,d\kappa$, and converges to it by Constant Sequences and Index-Shifted Sequences of Real Numbers §constant.

Claim (tight subfamilies). A subset of a tight set of Borel measures on $(Y,d)$ is tight, since for each $\varepsilon>0$ the compact set $K$ supplied for the larger set by Tight Family of Borel Measures on a Metric Space §tight serves for the subset; in particular a sequence whose terms all lie in a tight set is tight in the sense of Tight Family of Borel Measures on a Metric Space §sequence.

Claim (Prokhorov on pairs). $(X\times X,d)$ is a metric space by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pairs, and the members of $\mathcal{P}(X\times X)$ are Borel measures of total mass $1$ on it by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §measures, with weak convergence that of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §weak. Hence Prokhorov's Theorem on a Metric Space: a Tight Sequence of Borel Probability Measures Has a Weakly Convergent Subsequence §subsequence applies to every tight sequence in $\mathcal{P}(X\times X)$: it has a subsequence converging weakly to a member of $\mathcal{P}(X\times X)$.

Step 1 (claim 1). By Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §displacement, applied with $\rho$ in place of $\mu$, the coupling $(\mathrm{id}_{X},\mathrm{id}_{X})_{\#}\rho$ belongs to $\Pi^{a}(\rho,\rho)$, so $(\rho,\rho)$ is noise-connected by Couplings of Finite Noise Cost and Their Noise Cost §connected, and $\rho\in\mathcal{P}^{a}_{\rho}$ by The Measures Noise-Connected to the Reference Measure §space. This is the clause reference. For the clause moments, let $\mu\in\mathcal{P}^{a}_{\rho}$. Then $\Pi^{a}(\mu,\rho)$ is nonempty by The Measures Noise-Connected to the Reference Measure §space and Couplings of Finite Noise Cost and Their Noise Cost §connected, and $\rho\in\mathcal{P}_{2}(X)$ by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §reference; by the last sentence of Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §support-bound, applied to the pair $(\mu,\rho)$, also $\mu\in\mathcal{P}_{2}(X)$.

Step 2 (claim 2). Let $\mu,\nu\in\mathcal{P}^{a}_{\rho}$. By The Measures Noise-Connected to the Reference Measure §space choose first $\gamma_{1}\in\Pi^{a}(\mu,\rho)$ and then $\gamma_{2}\in\Pi^{a}(\nu,\rho)$. By Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §swap, applied to the pair $(\nu,\rho)$, $\sigma_{\#}\gamma_{2}\in\Pi^{a}(\rho,\nu)$. The measures $\mu,\rho,\nu$ belong to $\mathcal{P}_{2}(X)$ by Step 1 and Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §reference. Apply Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs with its $\mu,\lambda,\nu$ taken to be $\mu,\rho,\nu$ and its $\pi_{12},\pi_{23}$ taken to be $\gamma_{1},\sigma_{\#}\gamma_{2}$: by Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §glued there is a gluing $\theta$ of $\gamma_{1}$ and $\sigma_{\#}\gamma_{2}$, and by Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §noise-triangle $(q_{1},q_{3})_{\#}\theta\in\Pi^{a}(\mu,\nu)$. So $(\mu,\nu)$ is noise-connected, and $W_{a}(\mu,\nu)$ is defined by The Noise Wasserstein Distance §distance. Nothing here requires $\mu\neq\nu$.

Step 3 (claim 3). Let $\mu,\nu\in\mathcal{P}(X)$ with $(\mu,\nu)$ noise-connected, and write $w=W_{a}(\mu,\nu)^{2}$, the greatest lower bound of the nonempty set $S=\{I^{a}(\gamma):\gamma\in\Pi^{a}(\mu,\nu)\}$, which is bounded below by $0$. For each $j\in\mathbb{N}$, by Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ §epsilon-below with $\varepsilon=1/j$, there is $\gamma\in\Pi^{a}(\mu,\nu)$ with $I^{a}(\gamma)<w+1/j$; choose one such coupling $\gamma_{j}$ for each $j$ (a countable sequence of choices, made in the order of $j$). Then $0\le I^{a}(\gamma_{j})<w+1$ for all $j$, so the sequence $(I^{a}(\gamma_{j}))_{j}$ is bounded, with bound $w+1>0$.

The set $\Pi(\mu,\nu)$ is tight in $X\times X$ by the last sentence of Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost §tight, and every $\gamma_{j}$ lies in it, so $(\gamma_{j})_{j}$ is tight by Claim (tight subfamilies). By Claim (Prokhorov on pairs) there are a strictly increasing $(j_{i})_{i\in\mathbb{N}}$ and $\gamma\in\mathcal{P}(X\times X)$ with $\gamma_{j_{i}}\Rightarrow\gamma$. The constant sequences with terms $\mu$ and $\nu$ converge weakly to $\mu$ and $\nu$ by Claim (constant sequences), and $\gamma_{j_{i}}\in\Pi^{a}(\mu,\nu)$ for every $i$, with $(I^{a}(\gamma_{j_{i}}))_{i}$ bounded by $w+1$. By Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §lsc, applied with all $\mu_{i}=\mu$, all $\nu_{i}=\nu$ and the couplings $\gamma_{j_{i}}$, we get $\gamma\in\Pi^{a}(\mu,\nu)$ and $I^{a}(\gamma)\le\liminf_{i}I^{a}(\gamma_{j_{i}})$.

We show that this limit inferior is at most $w$. Write $b_{i}=I^{a}(\gamma_{j_{i}})$ and $B_{k}=\{b_{i}:i\ge k\}$, as in Limit Inferior of a Bounded Sequence of Real Numbers. Since $j_{i}\ge i$ by Strictly Increasing Sequences of Natural Numbers Dominate Their Index, $b_{i}<w+1/j_{i}\le w+1/i$ for every $i$. Fix $k\in\mathbb{N}$ and $\varepsilon>0$; by the Archimedean property choose $i\ge k$ with $1/i<\varepsilon$; then $\inf B_{k}\le b_{i}<w+\varepsilon$. As $\varepsilon>0$ was arbitrary, $\inf B_{k}\le w$, for every $k$. Thus $w$ is an upper bound of $\{\inf B_{k}:k\in\mathbb{N}\}$, and $\liminf_{i}b_{i}\le w$ by Limit Inferior of a Bounded Sequence of Real Numbers. Hence $I^{a}(\gamma)\le w=W_{a}(\mu,\nu)^{2}\le I^{a}(\gamma)$, the last inequality by The Noise Wasserstein Distance §distance. So $I^{a}(\gamma)=W_{a}(\mu,\nu)^{2}$, and $\gamma$ is noise-optimal by Noise-Optimal Couplings §optimal.

Step 4 (claim 8). This step uses only claims 1 to 3. Let $\mu,\nu\in\mathcal{P}^{a}_{\rho}$. They belong to $\mathcal{P}_{2}(X)$ by Step 1, and $(\mu,\nu)$ is noise-connected by Step 2; by Step 3 choose a noise-optimal $\gamma\in\Pi^{a}(\mu,\nu)$, so $I^{a}(\gamma)=W_{a}(\mu,\nu)^{2}$. Since $\gamma\in\Pi(\mu,\nu)$, The Quadratic Wasserstein Distance on a Hilbert Space §distance gives $W_{2}(\mu,\nu)^{2}\le I(\gamma)$, and Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §support-bound gives $I(\gamma)\le\bar{a}\,I^{a}(\gamma)=\bar{a}\,W_{a}(\mu,\nu)^{2}$. As $\bar{a}>0$ by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §weights and $W_{2}(\mu,\nu),W_{a}(\mu,\nu)\ge0$, taking square roots gives $W_{2}(\mu,\nu)\le\sqrt{\bar{a}}\,W_{a}(\mu,\nu)$.

Step 5 (claim 4). Let $\mu,\nu\in\mathcal{P}^{a}_{\rho}$; both ordered pairs $(\mu,\nu)$ and $(\nu,\mu)$ are noise-connected by Step 2. Write $S_{\mu\nu}=\{I^{a}(\gamma):\gamma\in\Pi^{a}(\mu,\nu)\}$ and $S_{\nu\mu}=\{I^{a}(\gamma):\gamma\in\Pi^{a}(\nu,\mu)\}$. If $\gamma\in\Pi^{a}(\mu,\nu)$, then $\sigma_{\#}\gamma\in\Pi^{a}(\nu,\mu)$ and $I^{a}(\sigma_{\#}\gamma)=I^{a}(\gamma)$ by Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §swap, so $S_{\mu\nu}\subseteq S_{\nu\mu}$; the same clause applied to the pair $(\nu,\mu)$ gives $S_{\nu\mu}\subseteq S_{\mu\nu}$. Hence the two sets are equal, so are their greatest lower bounds, and so are the square roots: $W_{a}(\mu,\nu)=W_{a}(\nu,\mu)$ by The Noise Wasserstein Distance §distance.

Step 6 (claim 5). Let $\mu,\nu\in\mathcal{P}^{a}_{\rho}$. If $\mu=\nu$, then by Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §displacement the coupling $(\mathrm{id}_{X},\mathrm{id}_{X})_{\#}\mu\in\Pi^{a}(\mu,\mu)$ has noise cost $0$, so $0\le W_{a}(\mu,\mu)^{2}\le0$ and $W_{a}(\mu,\nu)=0$. Conversely, if $W_{a}(\mu,\nu)=0$, then Step 4 gives $0\le W_{2}(\mu,\nu)\le\sqrt{\bar{a}}\cdot0=0$, so $W_{2}(\mu,\nu)=0$; as $\mu,\nu\in\mathcal{P}_{2}(X)$ by Step 1, The Quadratic Wasserstein Distance is a Metric on the Probability Measures with Finite Second Moment on a Hilbert Space §separation gives $\mu=\nu$.

Step 7 (claim 6). Let $\mu,\nu,\lambda\in\mathcal{P}^{a}_{\rho}$. The pairs $(\mu,\nu)$, $(\nu,\lambda)$ and $(\mu,\lambda)$ are noise-connected by Step 2. By Step 3 choose first a noise-optimal $\gamma_{1}\in\Pi^{a}(\mu,\nu)$ and then a noise-optimal $\gamma_{2}\in\Pi^{a}(\nu,\lambda)$, so that $I^{a}(\gamma_{1})=W_{a}(\mu,\nu)^{2}$ and $I^{a}(\gamma_{2})=W_{a}(\nu,\lambda)^{2}$. The three measures lie in $\mathcal{P}_{2}(X)$ by Step 1. Apply Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs with its $\mu,\lambda,\nu$ taken to be $\mu,\nu,\lambda$ and its $\pi_{12},\pi_{23}$ taken to be $\gamma_{1},\gamma_{2}$: by Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §glued there is a gluing $\theta$ of $\gamma_{1}$ and $\gamma_{2}$, and by Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §noise-triangle $\gamma_{3}=(q_{1},q_{3})_{\#}\theta\in\Pi^{a}(\mu,\lambda)$ with $\sqrt{I^{a}(\gamma_{3})}\le\sqrt{I^{a}(\gamma_{1})}+\sqrt{I^{a}(\gamma_{2})}=W_{a}(\mu,\nu)+W_{a}(\nu,\lambda)$. Since $W_{a}(\mu,\lambda)^{2}\le I^{a}(\gamma_{3})$ by The Noise Wasserstein Distance §distance, taking square roots gives $W_{a}(\mu,\lambda)\le\sqrt{I^{a}(\gamma_{3})}\le W_{a}(\mu,\nu)+W_{a}(\nu,\lambda)$.

Step 8 (claim 7). By Step 2 and The Noise Wasserstein Distance §distance, $(\mu,\nu)\mapsto W_{a}(\mu,\nu)$ is a function from $\mathcal{P}^{a}_{\rho}\times\mathcal{P}^{a}_{\rho}$ to $\mathbb{R}$ with nonnegative values, which is condition 1 of Metric Space. Conditions 2, 3 and 4 of that definition are Steps 6, 5 and 7 respectively (with the three points of condition 4 named $\mu,\nu,\lambda$ in that order). Hence this restriction of $W_{a}$ is a metric on $\mathcal{P}^{a}_{\rho}$.

Step 9 (claim 9, first part). Let $\mu_{j},\nu_{j},\mu,\nu$ be as in claim 9, and let $M>0$ satisfy $|W_{a}(\mu_{j},\nu_{j})|\le M$ for all $j$. Write $a_{j}=W_{a}(\mu_{j},\nu_{j})$, defined by Step 2, and $L=\liminf_{j}a_{j}$. Since $0\le a_{j}$ for all $j$, $0\le L$ by claim 2 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence. For each $j$ choose, by Steps 2 and 3, a noise-optimal $\gamma_{j}\in\Pi^{a}(\mu_{j},\nu_{j})$, so $I^{a}(\gamma_{j})=a_{j}^{2}$ (a countable sequence of choices, in the order of $j$).

The sequences $(\mu_{j})_{j}$ and $(\nu_{j})_{j}$ converge weakly, hence are tight by Cauchy Sequences in the Quadratic Wasserstein Space and Weakly Convergent Sequences on a Hilbert Space are Tight §weakly-convergent; that is, the sets $\mathcal{M}=\{\mu_{j}:j\in\mathbb{N}\}$ and $\mathcal{N}=\{\nu_{j}:j\in\mathbb{N}\}$ are tight. By Couplings on a Hilbert Space: Tightness, Closedness under Weak Convergence, and Lower Semicontinuity of the Quadratic Cost §tight the set $\mathcal{F}$ of all couplings of a member of $\mathcal{M}$ with a member of $\mathcal{N}$ is tight in $X\times X$, and every $\gamma_{j}$ belongs to $\mathcal{F}$.

Fix $\varepsilon>0$. By claim 4 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence there are natural numbers $n_{1}<n_{2}<\cdots$ with $a_{n_{l}}<L+\varepsilon$ for every $l$. The sequence $(\gamma_{n_{l}})_{l}$ has all its terms in $\mathcal{F}$, so it is tight by Claim (tight subfamilies), and by Claim (Prokhorov on pairs) there are a strictly increasing $(l_{i})_{i\in\mathbb{N}}$ and $\gamma\in\mathcal{P}(X\times X)$ with $\gamma_{n_{l_{i}}}\Rightarrow\gamma$. Put $k_{i}=n_{l_{i}}$; $(k_{i})_{i}$ is strictly increasing by claim 2 of A Subsequence of a Subsequence is a Subsequence, so $\mu_{k_{i}}\Rightarrow\mu$ and $\nu_{k_{i}}\Rightarrow\nu$ by Claim (subsequences). Moreover $0\le I^{a}(\gamma_{k_{i}})=a_{k_{i}}^{2}\le M^{2}$, so $(I^{a}(\gamma_{k_{i}}))_{i}$ is bounded, and $I^{a}(\gamma_{k_{i}})<(L+\varepsilon)^{2}$ for every $i$, because $0\le a_{k_{i}}<L+\varepsilon$. By Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §lsc, applied to the sequences $(\mu_{k_{i}})_{i}$, $(\nu_{k_{i}})_{i}$ and $(\gamma_{k_{i}})_{i}$, we get $\gamma\in\Pi^{a}(\mu,\nu)$ and $I^{a}(\gamma)\le\liminf_{i}I^{a}(\gamma_{k_{i}})$; and $\liminf_{i}I^{a}(\gamma_{k_{i}})\le\limsup_{i}I^{a}(\gamma_{k_{i}})\le(L+\varepsilon)^{2}$ by claims 1 and 2 of Basic Properties of the Limit Inferior and Limit Superior of a Bounded Real Sequence.

Consequently $\Pi^{a}(\mu,\nu)$ is nonempty and $(\mu,\nu)$ is noise-connected by Couplings of Finite Noise Cost and Their Noise Cost §connected, and $W_{a}(\mu,\nu)^{2}\le I^{a}(\gamma)\le(L+\varepsilon)^{2}$ by The Noise Wasserstein Distance §distance; taking square roots, $W_{a}(\mu,\nu)\le L+\varepsilon$. As $\varepsilon>0$ was arbitrary, $W_{a}(\mu,\nu)\le L=\liminf_{j}W_{a}(\mu_{j},\nu_{j})$.

Step 10 (claim 9, clause closed-under-limit). Suppose moreover $\mu\in\mathcal{P}^{a}_{\rho}$. By Step 9 choose $\gamma\in\Pi^{a}(\mu,\nu)$, and by The Measures Noise-Connected to the Reference Measure §space choose then $\gamma'\in\Pi^{a}(\mu,\rho)$. Since $\mu\in\mathcal{P}_{2}(X)$ by Step 1, the last sentence of Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §support-bound, applied to the pair $(\mu,\nu)$, gives $\nu\in\mathcal{P}_{2}(X)$; and $\rho\in\mathcal{P}_{2}(X)$ by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §reference. By Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §swap, $\sigma_{\#}\gamma'\in\Pi^{a}(\rho,\mu)$. Apply Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs with its $\mu,\lambda,\nu$ taken to be $\rho,\mu,\nu$ and its $\pi_{12},\pi_{23}$ taken to be $\sigma_{\#}\gamma',\gamma$: by Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §glued there is a gluing $\theta$, and by Gluing Two Couplings on a Hilbert Space over a Common Middle Marginal, and the Triangle Inequalities for the Quadratic and Noise Costs §noise-triangle $(q_{1},q_{3})_{\#}\theta\in\Pi^{a}(\rho,\nu)$. By Couplings of Finite Noise Cost: the Support Bound, Swap, Displacement Couplings and Lower Semicontinuity of the Noise Cost §swap once more, $\sigma_{\#}(q_{1},q_{3})_{\#}\theta\in\Pi^{a}(\nu,\rho)$, so $(\nu,\rho)$ is noise-connected and $\nu\in\mathcal{P}^{a}_{\rho}$ by The Measures Noise-Connected to the Reference Measure §space.

All nine claims are proved.