Proof of Stability of the Optimal Map and of the Displacement on the Torus Along Couplings of Nearly Optimal Cost

Throughout, each result cited is universally quantified over the data appearing in its own statement and is applied to the data named here. Every integral below is formed in a measure space named explicitly, namely $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\mu)$ or $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\gamma)$ for the probability measure $\gamma$ at hand, as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures; the cell measure space of The Flat Torus: Standing Notation §measure plays no role. Let $\iota_{\mathbb{R}}$ be the canonical map from $\mathbb{N}$ to $\mathbb{R}$, positive by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. Write $W=W_{\mathbb{T}}(\mu,\nu)$. The metric of $\mathbb{R}^{d}$ and of $\mathbb{R}^{d+d}$ is the Euclidean distance $d_{E}$, with $d_{E}(a,b)=\lVert a-b\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, and $\mathbb{R}$ carries the absolute-value metric.

Preliminary 1 (The functions $G_{S}$). For a Borel map $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ put

$$G_{S}(w)=d_{\mathbb{T}}\bigl(S(\mathrm{pr}_{1}(w)),\mathrm{pr}_{2}(w)\bigr)^{2}=\bigl\lVert\varpi\bigl(\mathrm{pr}_{2}(w)-S(\mathrm{pr}_{1}(w))\bigr)\bigr\rVert^{2}\qquad(w\in\mathbb{R}^{d+d}),$$

the two expressions agreeing by The Wrapped Displacement and the Flat Torus Distance §distance. The map $S\circ\mathrm{pr}_{1}$ is Borel as a composite of Borel maps, by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, the projections being Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections; the map $w\mapsto\mathrm{pr}_{2}(w)-S(\mathrm{pr}_{1}(w))$ is Borel because its components are differences of Borel real functions (claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets and claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); its composite with $\varpi$, Borel by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz, is Borel; and so $G_{S}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions. By The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, $0\le G_{S}\le d/4$. Hence $G_{S}$ is integrable against every $\gamma\in\mathcal{P}(\mathbb{R}^{d+d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, and by claim 6(c) of Borel Measurability and Bounded Integration on a Metric Space its integral as a nonnegative function is the same real number; so claim 1 of Linearity and Monotonicity of the Lebesgue Integral (additivity, nonnegative multiples, monotonicity) applies to it. For $S=\mathrm{id}$, $G_{\mathrm{id}}$ is the integrand of the torus cost, so $I_{\mathbb{T}}(\gamma)=\int G_{\mathrm{id}}\,d\gamma$ for $\gamma\in\Pi(\mu,\nu)$; and $G_{T}$ is the first function of the statement.

Preliminary 2 (Continuity of $G_{S}$ for Lipschitz $S$). Let $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be Lipschitz with constant $\Lambda$, a nonnegative real; it is continuous by A Lipschitz Map is Uniformly Continuous, hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space together with Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces. Let $w,w'\in\mathbb{R}^{d+d}$ and write $x=\mathrm{pr}_{1}(w)$, $y=\mathrm{pr}_{2}(w)$, $x'=\mathrm{pr}_{1}(w')$, $y'=\mathrm{pr}_{2}(w')$. The second inequality of The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §lipschitz, applied to the points $S(x)$, $S(x')$ and $y$, gives

$$\bigl|d_{\mathbb{T}}(S(x),y)^{2}-d_{\mathbb{T}}(S(x'),y)^{2}\bigr|\le\sqrt{d}\,\lVert S(x)-S(x')\rVert\le\sqrt{d}\,\Lambda\,\lVert x-x'\rVert .$$

By The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §symmetry, $d_{\mathbb{T}}(S(x'),y)=d_{\mathbb{T}}(y,S(x'))$ and $d_{\mathbb{T}}(S(x'),y')=d_{\mathbb{T}}(y',S(x'))$, so the same inequality, applied to the points $y$, $y'$ and $S(x')$, gives

$$\bigl|d_{\mathbb{T}}(S(x'),y)^{2}-d_{\mathbb{T}}(S(x'),y')^{2}\bigr|\le\sqrt{d}\,\lVert y-y'\rVert .$$

The projections are Lipschitz with constant $1$ by claim 1 of Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit, so $\lVert x-x'\rVert\le\lVert w-w'\rVert$ and $\lVert y-y'\rVert\le\lVert w-w'\rVert$. Adding the two displays with the triangle inequality for the absolute value,

$$|G_{S}(w)-G_{S}(w')|\le\sqrt{d}\,(\Lambda+1)\,\lVert w-w'\rVert .$$

Thus $G_{S}$ is Lipschitz with constant $\sqrt{d}(\Lambda+1)$ from $(\mathbb{R}^{d+d},d_{E})$ to $\mathbb{R}$, hence continuous by A Lipschitz Map is Uniformly Continuous, and it is bounded by Preliminary 1. The identity map of $\mathbb{R}^{d}$ is Lipschitz with constant $1$, so in particular $G_{\mathrm{id}}$ is bounded and continuous.

Step 1 (Comparison of two maps). Let $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$ be Borel and let $\gamma\in\Pi(\mu,\nu)$. For $w\in\mathbb{R}^{d+d}$ with $x=\mathrm{pr}_{1}(w)$ and $y=\mathrm{pr}_{2}(w)$, The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §triangle and the last assertion of The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §minimal give

$$d_{\mathbb{T}}(T(x),y)\le d_{\mathbb{T}}(T(x),S(x))+d_{\mathbb{T}}(S(x),y)\le\lVert T(x)-S(x)\rVert+d_{\mathbb{T}}(S(x),y).$$

Both sides are nonnegative, squaring preserves the order of nonnegative reals, and $(a+b)^{2}\le2a^{2}+2b^{2}$ for real $a,b$ because $2a^{2}+2b^{2}-(a+b)^{2}=(a-b)^{2}\ge0$; hence

$$G_{T}(w)\le2\,\lVert T(\mathrm{pr}_{1}(w))-S(\mathrm{pr}_{1}(w))\rVert^{2}+2\,G_{S}(w).$$

The first function on the right is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, applied to the Borel maps $T\circ\mathrm{pr}_{1}$ and $S\circ\mathrm{pr}_{1}$. Integrating against $\gamma$ with claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and using the change-of-variables formula with $(\mathrm{pr}_{1})_{\#}\gamma=\mu$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling),

$$\int_{\mathbb{R}^{d+d}}G_{T}\,d\gamma\le2\int_{\mathbb{R}^{d}}\lVert T-S\rVert^{2}\,d\mu+2\int_{\mathbb{R}^{d+d}}G_{S}\,d\gamma . \tag{$**$}$$

Step 2 (The limit of a weakly convergent subsequence is the coupling induced by $T$). Let $(\sigma_{k})_{k\in\mathbb{N}}$ be a sequence in $\Pi(\mu,\nu)$ such that $(I_{\mathbb{T}}(\sigma_{k}))_{k}$ converges to $W^{2}$. By The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions §inclusion, $\mu$ and $\nu$ belong to $\mathcal{P}_{2}(\mathbb{R}^{d})$, so by Tightness from Bounded Second Moments, and Tightness of the Couplings of Two Measures with Finite Second Moment §couplings, applied with $m=d$, the set $\Pi(\mu,\nu)$ is tight in $(\mathbb{R}^{d+d},d_{E})$. The set of terms of $(\sigma_{k})_{k}$ is a subset of $\Pi(\mu,\nu)$, and a compact set that serves $\Pi(\mu,\nu)$ for a given positive $\varepsilon$ serves every subset of it; so the sequence is tight in the sense of Tight Family of Borel Measures on a Metric Space §sequence. By Prokhorov's Theorem on Euclidean Space: a Tight Sequence of Probability Measures Has a Weakly Convergent Subsequence, applied with $m=d+d$, there are a strictly increasing sequence $(k_{j})_{j\in\mathbb{N}}$ and $\pi\in\mathcal{P}(\mathbb{R}^{d+d})$ such that $(\sigma_{k_{j}})_{j}$ converges weakly to $\pi$ on $(\mathbb{R}^{d+d},d_{E})$, and $\pi\in\Pi(\mu,\nu)$ by The Couplings of Two Probability Measures on Euclidean Space are Closed under Weak Convergence, applied with $m=d$.

Since $G_{\mathrm{id}}$ is bounded and continuous (Preliminary 2), Weak Convergence of Finite Borel Measures on a Metric Space shows that $(I_{\mathbb{T}}(\sigma_{k_{j}}))_{j}$ converges to $I_{\mathbb{T}}(\pi)$; and it converges to $W^{2}$ by A Subsequence of a Convergent Sequence Has the Same Limit. Given a positive real $\eta$, pick $j$ beyond both thresholds furnished by Limit of a Sequence of Real Numbers for $\eta$; then $|I_{\mathbb{T}}(\sigma_{k_{j}})-I_{\mathbb{T}}(\pi)|<\eta$ and $|I_{\mathbb{T}}(\sigma_{k_{j}})-W^{2}|<\eta$, so $I_{\mathbb{T}}(\pi)<W^{2}+2\eta$. Were $W^{2}<I_{\mathbb{T}}(\pi)$, the choice $\eta=\tfrac12(I_{\mathbb{T}}(\pi)-W^{2})$ would give a contradiction; hence $I_{\mathbb{T}}(\pi)\le W^{2}$. Conversely, by Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings §distance $W$ is the nonnegative square root of the greatest lower bound of $\{I_{\mathbb{T}}(\gamma):\gamma\in\Pi(\mu,\nu)\}$, so $W^{2}$ is that greatest lower bound and $W^{2}\le I_{\mathbb{T}}(\pi)$. Therefore $I_{\mathbb{T}}(\pi)=W^{2}$ and $\pi$ is optimal. The coupling $(\mathrm{id},T)_{\#}\mu$ is optimal as well, $T$ being an optimal map on the torus. As $(\mu,\nu)$ is uniquely mapped on the torus, there is an optimal map $T_{0}$ such that every optimal coupling of $\mu$ and $\nu$ equals $(\mathrm{id},T_{0})_{\#}\mu$; applying this to $\pi$ and to $(\mathrm{id},T)_{\#}\mu$ gives $\pi=(\mathrm{id},T)_{\#}\mu$.

Step 3 (The subsequence has vanishing discrepancy). Keep the notation of Step 2 and let $\eta$ be a positive real number. The map $T$ is Borel and, by the change-of-variables formula applied to the Borel function $\lVert\cdot\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions and to $T_{\#}\mu=\nu$, satisfies $\int_{\mathbb{R}^{d}}\lVert T\rVert^{2}\,d\mu=M_{2}(\nu)\le d$, by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment and The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions §inclusion. So its class lies in the space $L^{2}(\mu;\mathbb{R}^{d})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu. Put $\eta'=\min(\eta,1)$, a positive real number. By The Bounded Lipschitz Vector Fields are Dense in the Square-Integrable Vector Fields Against a Probability Measure §dense, applied with $\xi=T$ and slack $\eta'$, there is a bounded Lipschitz map $S:\mathbb{R}^{d}\to\mathbb{R}^{d}$, Lipschitz with some constant $\Lambda$, with $\lVert T-S\rVert_{\mu}\le\eta'$; by the formula for $\lVert\cdot\rVert_{\mu}$ in Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, and as $0<\eta'\le1$,

$$\int_{\mathbb{R}^{d}}\lVert T-S\rVert^{2}\,d\mu=\lVert T-S\rVert_{\mu}^{2}\le\eta'^{2}\le\eta'\le\eta .$$

By Preliminary 2, $G_{S}$ is bounded and continuous.

The pairing $(\mathrm{id},T)$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing, and $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ send its value at $x$ to $x$ and $T(x)$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections. Using $\pi=(\mathrm{id},T)_{\#}\mu$ (Step 2) and the change-of-variables formula, then The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §symmetry and the last assertion of The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §minimal (squared, both sides being nonnegative) with claim 1 of Linearity and Monotonicity of the Lebesgue Integral,

$$\int_{\mathbb{R}^{d+d}}G_{S}\,d\pi=\int_{\mathbb{R}^{d}}d_{\mathbb{T}}(S(x),T(x))^{2}\,\mu(dx)=\int_{\mathbb{R}^{d}}d_{\mathbb{T}}(T(x),S(x))^{2}\,\mu(dx)\le\int_{\mathbb{R}^{d}}\lVert T-S\rVert^{2}\,d\mu\le\eta .$$

Since $G_{S}$ is bounded and continuous and $(\sigma_{k_{j}})_{j}$ converges weakly to $\pi$, the sequence $\bigl(\int G_{S}\,d\sigma_{k_{j}}\bigr)_{j}$ converges to $\int G_{S}\,d\pi$, so by Limit of a Sequence of Real Numbers, used with $\eta$, there is $j_{2}\in\mathbb{N}$ with $\int G_{S}\,d\sigma_{k_{j}}<\int G_{S}\,d\pi+\eta\le2\eta$ for every $j\ge j_{2}$. The order of choice is $\eta$, then $S$ and $\Lambda$, then $j_{2}$. Applying $(**)$ with $\gamma=\sigma_{k_{j}}$,

$$\int_{\mathbb{R}^{d+d}}G_{T}\,d\sigma_{k_{j}}\le2\eta+2\cdot2\eta=6\eta\qquad\text{for every }j\ge j_{2}.$$

Step 4 (Claim 1). Put $a_{n}=\int G_{T}\,d\gamma_{n}$, a nonnegative real number by Preliminary 1. Suppose that $(a_{n})_{n\in\mathbb{N}}$ does not converge to $0$. By Limit of a Sequence of Real Numbers there is a positive real $\varepsilon$ such that for every $N\in\mathbb{N}$ some $n\ge N$ has $|a_{n}-0|\ge\varepsilon$, that is $a_{n}\ge\varepsilon$. Choose $n_{1}$ with $a_{n_{1}}\ge\varepsilon$ and, recursively, given $n_{k}$, choose $n_{k+1}\ge n_{k}+1$ with $a_{n_{k+1}}\ge\varepsilon$; then $(n_{k})_{k}$ is strictly increasing and $(\sigma_{k})_{k}=(\gamma_{n_{k}})_{k}$ is a subsequence of $(\gamma_{n})_{n}$ in $\Pi(\mu,\nu)$, whose costs $I_{\mathbb{T}}(\sigma_{k})$ converge to $W^{2}$ by A Subsequence of a Convergent Sequence Has the Same Limit. Steps 2 and 3 apply to it; with $\eta=\varepsilon/12$ they yield a strictly increasing $(k_{j})_{j}$ and $j_{2}$ with $a_{n_{k_{j_{2}}}}=\int G_{T}\,d\sigma_{k_{j_{2}}}\le\varepsilon/2<\varepsilon$, contradicting $a_{n_{k_{j_{2}}}}\ge\varepsilon$. Hence $(a_{n})_{n}$ converges to $0$, which is claim 1.

Step 5 (Claim 2: a pointwise comparison). Assume now that $\mu$ is absolutely continuous. By McCann's Theorem on the Flat Torus: Optimal Couplings out of an Absolutely Continuous Measure are Induced by a Unique Map with a Periodic Potential §potential, applied to the optimal map $T$ on the torus from $\mu$ to $\nu$, there is $D\in\mathcal{B}(\mathbb{R}^{d})$ with $\mu(D)=1$ such that $T(x)-x$ is regular for every $x\in D$. Let $r$ be the function of The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §regular and put $\rho(x)=r(T(x)-x)$ for $x\in\mathbb{R}^{d}$. The map $\rho$ is Borel: $r$ is Borel by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §regular, $T-\mathrm{id}$ is Borel as recorded in Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §displacement, and composites of Borel maps are Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. By The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §regular, $\rho(x)>0$ for every $x\in D$.

Write $F(w)=\lVert\varpi(\mathrm{pr}_{2}(w)-\mathrm{pr}_{1}(w))-v_{T}(\mathrm{pr}_{1}(w))\rVert^{2}$ for $w\in\mathbb{R}^{d+d}$. It is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, applied to the Borel maps $\varpi\circ(\mathrm{pr}_{2}-\mathrm{pr}_{1})$ (Borel as in Preliminary 1) and $v_{T}\circ\mathrm{pr}_{1}$ ($v_{T}$ being Borel by Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §displacement). Fix $w$ and write $x=\mathrm{pr}_{1}(w)$, $y=\mathrm{pr}_{2}(w)$.

(a) $F(w)\le d$: by the first inequality of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, $F(w)\le2\lVert\varpi(y-x)\rVert^{2}+2\lVert\varpi(T(x)-x)\rVert^{2}$, recalling $v_{T}(x)=\varpi(T(x)-x)$ from Optimal Maps on the Torus, Uniquely Mapped Pairs and the Displacement Field of a Map §displacement, and each squared norm is at most $d/4$ by The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range.

(b) If $x\in D$ and $d_{\mathbb{T}}(T(x),y)<\rho(x)$, then $F(w)=G_{T}(w)$. Indeed, put $z=T(x)-x$ and $e=\varpi(y-T(x))$. By The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range, $k=(y-T(x))-e$ lies in $\mathbb{Z}^{d}$, and $\lVert e\rVert=d_{\mathbb{T}}(T(x),y)$ by The Wrapped Displacement and the Flat Torus Distance §distance. Then $y-x=(z+e)+k$, so $\varpi(y-x)=\varpi(z+e)$ by the periodicity in The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §range. As $x\in D$, $z$ is regular and $r(z)=\rho(x)$; since $\lVert(z+e)-z\rVert=\lVert e\rVert<r(z)$, The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound §regular gives $\varpi(z+e)=\varpi(z)+e=v_{T}(x)+e$. Hence $\varpi(y-x)-v_{T}(x)=e$ and $F(w)=\lVert e\rVert^{2}=d_{\mathbb{T}}(T(x),y)^{2}=G_{T}(w)$.

(c) Let $\eta$ be a positive real number and put $A_{\eta}=\{x\in D:\eta<\rho(x)\}$, which belongs to $\mathcal{B}(\mathbb{R}^{d})$ because $\{x:\eta<\rho(x)\}$ does by the criterion of Measure Spaces and the Lebesgue Integral: Standing Notation §measurable for the measurable space $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$; and put $B_{\eta}=\mathrm{pr}_{1}^{-1}(\mathbb{R}^{d}\setminus A_{\eta})$, which belongs to $\mathcal{B}(\mathbb{R}^{d+d})$ since $\mathrm{pr}_{1}$ is Borel. Then for every $w\in\mathbb{R}^{d+d}$

$$F(w)\le\bigl(1+d\,\eta^{-2}\bigr)\,G_{T}(w)+d\,\mathbf{1}_{B_{\eta}}(w). \tag{$\ddagger$}$$

All terms on the right are nonnegative. If $w\in B_{\eta}$, then $F(w)\le d=d\,\mathbf{1}_{B_{\eta}}(w)$ by (a). If $w\notin B_{\eta}$ and $\eta^{2}\le G_{T}(w)$, then $F(w)\le d\le d\,\eta^{-2}G_{T}(w)$ by (a). If $w\notin B_{\eta}$ and $G_{T}(w)<\eta^{2}$, then $x\in A_{\eta}$, so $x\in D$ and $\eta<\rho(x)$; moreover $d_{\mathbb{T}}(T(x),y)<\eta$, since $\eta\le d_{\mathbb{T}}(T(x),y)$ would give $\eta^{2}\le G_{T}(w)$; thus $d_{\mathbb{T}}(T(x),y)<\rho(x)$ and (b) gives $F(w)=G_{T}(w)$.

The right side of $(\ddagger)$ is Borel by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. Integrating $(\ddagger)$ against $\gamma_{j}$ with claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and using The Integral of an Indicator Function is the Measure of the Set together with $\gamma_{j}(\mathrm{pr}_{1}^{-1}(\mathbb{R}^{d}\setminus A_{\eta}))=\mu(\mathbb{R}^{d}\setminus A_{\eta})$ from Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling, we get for every $j\in\mathbb{N}$

$$0\le\int_{\mathbb{R}^{d+d}}F\,d\gamma_{j}\le\bigl(1+d\,\eta^{-2}\bigr)\int_{\mathbb{R}^{d+d}}G_{T}\,d\gamma_{j}+d\,\mu(\mathbb{R}^{d}\setminus A_{\eta}). \tag{$\S$}$$

Step 6 (The exceptional set is small). For $m\in\mathbb{N}$ put $\eta_{m}=\iota_{\mathbb{R}}(m)^{-1}$, a positive real by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and $A_{m}=A_{\eta_{m}}$. By claims 1 and 3 there, $0<\iota_{\mathbb{R}}(m)<\iota_{\mathbb{R}}(m)+1=\iota_{\mathbb{R}}(m+1)$, so $\eta_{m+1}\le\eta_{m}$ by Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares §reciprocal, applied with $a=\iota_{\mathbb{R}}(m)$ and $b=\iota_{\mathbb{R}}(m+1)$, and therefore $A_{m}\subseteq A_{m+1}$. Every $A_{m}$ is contained in $D$; conversely, for $x\in D$ one has $\rho(x)>0$, and claim 3 of The Archimedean Property of the Real Numbers gives $n\in\mathbb{N}$ with $0<\iota_{\mathbb{R}}(n)^{-1}<\rho(x)$, so $x\in A_{n}$. Hence $\bigcup_{m}A_{m}=D$. Each $\mu(A_{m})$ is a real number at most $\mu(\mathbb{R}^{d})=1$, by claim 2 of Basic Properties of a Measure, so claim 5 there (continuity from below) shows that $(\mu(A_{m}))_{m}$ converges to $\mu(D)=1$; and $\mu(\mathbb{R}^{d}\setminus A_{m})=1-\mu(A_{m})$ by claim 3 there. Hence $(\mu(\mathbb{R}^{d}\setminus A_{m}))_{m}$ converges to $0$.

Step 7 (Claim 2). Let $\varepsilon$ be a positive real number. By Step 6 choose $m\in\mathbb{N}$ with $\mu(\mathbb{R}^{d}\setminus A_{m})<\varepsilon/(2d)$. Put $\eta=\eta_{m}$ and $c=1+d\,\eta^{-2}$, a positive real. By claim 1 choose $N\in\mathbb{N}$ with $\int G_{T}\,d\gamma_{j}<\varepsilon/(2c)$ for every $j\ge N$. The order of choice is $\varepsilon$, then $m$, then $N$. For every $j\ge N$, $(\S)$ gives

$$\Bigl|\int_{\mathbb{R}^{d+d}}F\,d\gamma_{j}-0\Bigr|=\int_{\mathbb{R}^{d+d}}F\,d\gamma_{j}<c\cdot\frac{\varepsilon}{2c}+d\cdot\frac{\varepsilon}{2d}=\varepsilon .$$

By Limit of a Sequence of Real Numbers the sequence $\bigl(\int F\,d\gamma_{j}\bigr)_{j\in\mathbb{N}}$ converges to $0$, which is claim 2.