Proof of Displacement Convexity of the Logarithmic Energy on the Real Line

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named at the point of use. The rules of Elementary Order Arithmetic in an Ordered Field, Elementary Arithmetic in an Ordered Field and Properties of the Absolute Value in an Ordered Field for adding, multiplying and scaling inequalities and for absolute values are used without further mention, as are the rules for limits of sums, products and quotients of convergent real sequences and the fact that a non-strict inequality between the terms of convergent real sequences passes to their limits (limits being those of Limit of a Sequence of Real Numbers).

Conventions. As in One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative, a point of $\mathbb{R}^{2}$ is written $(x,y)$ with $x,y\in\mathbb{R}$, and the coordinate projections are $\mathrm{pr}_{1}(x,y)=x$ and $\mathrm{pr}_{2}(x,y)=y$, which are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections. For $A,B\in\mathcal{B}(\mathbb{R})$ we write $A\times B$ for the set $\{(x,y):x\in A,\ y\in B\}=\mathrm{pr}_{1}^{-1}(A)\cap\mathrm{pr}_{2}^{-1}(B)$; by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product it is Borel and $(\rho\boxtimes\rho')(A\times B)=\rho(A)\rho'(B)$ for $\rho,\rho'\in\mathcal{P}(\mathbb{R})$. $\lambda_{1}$ is the Lebesgue measure on $\mathcal{B}(\mathbb{R})$. Integrals against members of $\mathcal{P}(\mathbb{R})$ and $\mathcal{P}(\mathbb{R}^{2})$ are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, where it is recorded that a bounded Borel function is integrable against each of them. Null means of measure $0$ for the measure in question; by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §null-union finite and countable unions of null sets are null. Dominated convergence always refers to The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §dominated, and Fatou to Fatou's Lemma. A function $f:\mathbb{R}\to\mathbb{R}$ is Lipschitz with constant $L$, for a real $L\ge0$, if $|f(x)-f(y)|\le L|x-y|$ for all $x,y\in\mathbb{R}$, which is Lipschitz Map Between Metric Spaces for the absolute-value metric; such an $f$ is continuous by A Lipschitz Map is Uniformly Continuous, hence Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. Borel maps are closed under composition by the same clause.

Fix $\mu$, $\nu$ and $\pi$ as in the statement.

Step 0 (Standing facts).

(0-) Differentiability implies continuity. If $f:\mathbb{R}\to\mathbb{R}$ is differentiable at $x_{0}$ in the sense of Derivative at an Interior Point, then, taking $\varepsilon=1$ there, there is $\delta>0$ with $|f(x_{0}+h)-f(x_{0})|\le(|f'(x_{0})|+1)|h|$ whenever $0<|h|<\delta$; hence $f$ is continuous at $x_{0}$. A function differentiable at every point is therefore continuous, and so is its restriction to any closed interval.

(0a) Atomless measures and the diagonal. Since $\mu,\nu\in\mathcal{D}_{\log}$, by The Logarithmic Energy of a Probability Measure on the Real Line §energy both belong to $\mathcal{P}_{2}(\mathbb{R})$ and give measure $0$ to every one-point set, that is, both are atomless. The diagonal $\Delta=\{(x,x):x\in\mathbb{R}\}$ is Borel by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure §borel, and $(\mu\boxtimes\mu)(\Delta)=0=(\nu\boxtimes\nu)(\Delta)$ by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure §diagonal. If $A\in\mathcal{B}(\mathbb{R})$ and $\mu(A)=0$, then $A\times\mathbb{R}$ and $\mathbb{R}\times A$ are $\mu\boxtimes\mu$-null by the product formula recalled above.

(0b) The score. Fix a Borel map $\xi:\mathbb{R}\to\mathbb{R}$ representing $\Xi_{\mu}\in L^{2}(\mu;\mathbb{R})$; by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §scalars, $\int_{\mathbb{R}}\xi^{2}\,d\mu<\infty$ and $\langle\Xi_{\mu},\eta\rangle_{\mu}=\int_{\mathbb{R}}\xi\,\eta\,d\mu$ for every $\eta\in L^{2}(\mu;\mathbb{R})$. For every Borel $\eta:\mathbb{R}\to\mathbb{R}$ with $\int_{\mathbb{R}}\eta^{2}\,d\mu<\infty$ the product $\xi\eta$ is integrable, since $|\xi\eta|\le\tfrac12(\xi^{2}+\eta^{2})$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions (with $q=1$) and integrals of nonnegative functions are monotone and additive by claim 1 of Linearity and Monotonicity of the Lebesgue Integral. Taking $\eta=1$, $\xi$ is integrable; put $N_{\xi}=\int_{\mathbb{R}}|\xi|\,d\mu$.

(0c) Difference quotients. For $\varphi:\mathbb{R}\to\mathbb{R}$ let $Q_{\varphi}:\mathbb{R}^{2}\to\mathbb{R}$ be $Q_{\varphi}(x,y)=\frac{\varphi(x)-\varphi(y)}{x-y}$ if $x\ne y$ and $Q_{\varphi}(x,x)=0$. Clearly $Q_{\varphi}(x,y)=Q_{\varphi}(y,x)$, and $Q_{a\varphi+b\varphi'}=aQ_{\varphi}+bQ_{\varphi'}$ for $a,b\in\mathbb{R}$. If $\varphi$ is Borel, so is $Q_{\varphi}$. Indeed, let $\ell$ be the logarithmic kernel of The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure, Borel by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure §borel. The exponential function is differentiable at every point by claim 3 of Basic Properties of the Exponential Function, hence continuous by (0-), hence Borel, so $\exp\circ\ell$ is Borel. Let $w(x,y)=(x-y)\exp(\ell(x,y))^{2}$, Borel by claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. For $x\ne y$, $\exp(\ell(x,y))=\exp(-\log|x-y|)=1/\exp(\log|x-y|)=1/|x-y|$ by claim 2 of Basic Properties of the Exponential Function and The Natural Logarithm, so $w(x,y)=(x-y)/(x-y)^{2}=1/(x-y)$; and $w(x,x)=0$. Hence $Q_{\varphi}=(\varphi\circ\mathrm{pr}_{1}-\varphi\circ\mathrm{pr}_{2})\,w$, which is Borel by the same claims.

(0d) Bounds on quotients. If $\varphi$ is Lipschitz with constant $L$, then $|Q_{\varphi}|\le L$ everywhere, so $Q_{\varphi}$ is bounded and Borel, hence integrable against $\mu\boxtimes\mu$. If $\varphi$ is differentiable at every point with continuous derivative bounded in absolute value by $L$, then off $\Delta$ the function $Q_{\varphi}$ coincides with the function $F$ of The Difference Quotient of a Function with Bounded Continuous Derivative is Bounded, Symmetric and Continuous on the Plane, so $|Q_{\varphi}|\le L$ by The Difference Quotient of a Function with Bounded Continuous Derivative is Bounded, Symmetric and Continuous on the Plane §bound, and $\varphi$ is Lipschitz with constant $L$. In particular this applies to every $g\in C_{c}^{\infty}(\mathbb{R})$, whose derivative $g'$ is continuous and bounded by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives; such $g$ is continuous (being differentiable, by (0-)) and compactly supported, hence bounded by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable.

(0e) The score identity in terms of $Q$. Let $\psi\in C_{c}^{\infty}(\mathbb{R})$. The function $F_{\psi}$ of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §quotient equals $Q_{\psi'}$ off $\Delta$, and $\Delta$ is $\mu\boxtimes\mu$-null by (0a); both are integrable (by that clause and by (0d)), so $\int F_{\psi}\,d(\mu\boxtimes\mu)=\int Q_{\psi'}\,d(\mu\boxtimes\mu)$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison. Hence the defining identity of Finite Free Fisher Information, the Free Score and the Free Fisher Information of a Probability Measure on the Real Line §score, read through One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives, becomes

(0f) Kernels. Fix a mollifier kernel $\rho$ of radius $1$ on $\mathbb{R}=\mathbb{R}^{1}$, which exists by claim 2 of Existence of Mollifier Kernels of Every Radius. For a positive real $s$ let $\rho_{s}(y)=s^{-1}\rho(s^{-1}y)$; by Rescaling a Mollifier Kernel it is a mollifier kernel of radius $s$, so it is smooth, nonnegative, vanishes at every $y$ with $|y|>s$, and $\int_{\mathbb{R}}\rho_{s}\,d\lambda_{1}=1$. By claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set it is compactly supported, so $\rho_{s}\in C_{c}^{\infty}(\mathbb{R})$ by Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space. By (0d) there are reals $B_{\rho},L_{\rho}\ge0$ with $|\rho|\le B_{\rho}$ and $|Q_{\rho}|\le L_{\rho}$. Hence $|\rho_{s}|\le B_{\rho}/s$, and, since for $x\ne y$

also $|Q_{\rho_{s}}|\le L_{\rho}/s^{2}$ everywhere.

(0g) Mollifying a Lipschitz function. Let $f:\mathbb{R}\to\mathbb{R}$ be Lipschitz with constant $K$ and let $0<\varepsilon\le1$. Since $f$ is continuous and $\rho_{\varepsilon}$ is continuous and vanishes off $[-\varepsilon,\varepsilon]$, the convolution $f*\rho_{\varepsilon}$ is defined on all of $\mathbb{R}$ (the open set being $\mathbb{R}$), with $(f*\rho_{\varepsilon})(x)=\int_{\mathbb{R}}f(x-y)\rho_{\varepsilon}(y)\,\lambda_{1}(dy)$, the integrand being integrable by claim 1 of The Convolution Integrand is Continuous, Compactly Supported and Integrable. (g1) $f*\rho_{\varepsilon}$ is smooth by claim 2 of Convolution with a $C^k$ Kernel is of Class $C^k$. (g2) $|(f*\rho_{\varepsilon})(x)-f(x)|\le K\varepsilon$ for every $x$: as $\int\rho_{\varepsilon}\,d\lambda_{1}=1$, claim 2 of Linearity and Monotonicity of the Lebesgue Integral gives $(f*\rho_{\varepsilon})(x)-f(x)=\int(f(x-y)-f(x))\rho_{\varepsilon}(y)\,\lambda_{1}(dy)$, and the absolute value of this integrand is at most $K|y|\rho_{\varepsilon}(y)\le K\varepsilon\,\rho_{\varepsilon}(y)$ for every $y$, because $\rho_{\varepsilon}(y)=0$ when $|y|>\varepsilon$; the bound follows from the same claim. (g3) $f*\rho_{\varepsilon}$ is Lipschitz with constant $K$: $(f*\rho_{\varepsilon})(x)-(f*\rho_{\varepsilon})(x')=\int(f(x-y)-f(x'-y))\rho_{\varepsilon}(y)\,\lambda_{1}(dy)$ and the integrand is at most $K|x-x'|\rho_{\varepsilon}(y)$ in absolute value. (g4) If $A>0$ and $f(t)=0$ whenever $|t|>A$, then $(f*\rho_{\varepsilon})(x)=0$ whenever $|x|>A+1$, since for every $y$ either $|y|>\varepsilon$ and $\rho_{\varepsilon}(y)=0$, or $|y|\le\varepsilon\le1$ and $|x-y|\ge|x|-|y|>A$, so the integrand vanishes identically; then $f*\rho_{\varepsilon}$ is compactly supported by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set and belongs to $C_{c}^{\infty}(\mathbb{R})$ by Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space.

Step 1 (The score identity for every test function). For $g\in C_{c}^{\infty}(\mathbb{R})$ put

both integrals existing by (0b) and (0d), $g$ being bounded and Borel. We show $\kappa(g)=0$. Let $s>0$ and $c=\int_{\mathbb{R}}g\,d\lambda_{1}$, which exists by Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function §integrable. The function $\rho_{s}$ is a nonnegative test function of unit mass, as in claim 2 of Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function; so by Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function §correction the function $g-c\rho_{s}$ is a test function with $\lambda_{1}$-integral $0$, and by Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function §primitive there is $\psi\in C_{c}^{\infty}(\mathbb{R})$ with $\psi'=g-c\rho_{s}$. By (1), linearity of $Q$ and claim 2 of Linearity and Monotonicity of the Lebesgue Integral,

Applying (2) with $g=\rho_{t}$ (for which $c=1$) gives $\kappa(\rho_{t})=\kappa(\rho_{s})$ for all positive $s,t$. By (0f), for $s\ge1$,

So $|\kappa(\rho_{1})|\le(N_{\xi}B_{\rho}+L_{\rho})/s$ for every real $s\ge1$, whence $\kappa(\rho_{1})=0$, and (2) with $s=1$ gives $\kappa(g)=0$.

Step 2 (The score identity for Lipschitz functions). Let $\varphi:\mathbb{R}\to\mathbb{R}$ be Lipschitz with constant $L$. Then $\int_{\mathbb{R}}\varphi^{2}\,d\mu<\infty$, $\xi\varphi$ and $Q_{\varphi}$ are integrable, and

Since $|\varphi(x)|\le|\varphi(0)|+L|x|$, we have $\varphi(x)^{2}\le2\varphi(0)^{2}+2L^{2}x^{2}$, and $\int x^{2}\,\mu(dx)<\infty$ because $\mu\in\mathcal{P}_{2}(\mathbb{R})$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment §moment); so $\int\varphi^{2}\,d\mu<\infty$, and $\xi\varphi$ is integrable by (0b). $Q_{\varphi}$ is integrable by (0d).

For a real $R\ge1$ let $\chi_{R}$ be the cutoff of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff with $q=1$, and $M_{1}\ge0$ the constant there (independent of $R$): $\chi_{R}$ is smooth and compactly supported, $0\le\chi_{R}\le1$, $\chi_{R}(x)=1$ for $|x|\le R$, $\chi_{R}(x)=0$ for $|x|\ge2R$, and $|\partial_{1}\chi_{R}|\le M_{1}/R$. Thus $\chi_{R}\in C_{c}^{\infty}(\mathbb{R})$, its derivative is $\chi_{R}'=\partial_{1}\chi_{R}$ by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives, and by (0d) $|\chi_{R}(x)-\chi_{R}(y)|\le(M_{1}/R)|x-y|$. Put $f_{R}=\chi_{R}\varphi$ and $K=L+M_{1}|\varphi(0)|+2M_{1}L$.

(2a) $f_{R}$ is Lipschitz with constant $K$. Let $x,y\in\mathbb{R}$ with $|y|\le|x|$ (the other case follows by exchanging $x$ and $y$). If $|y|\ge2R$ then $|x|\ge2R$ and $f_{R}(x)=0=f_{R}(y)$. If $|y|<2R$, then $f_{R}(x)-f_{R}(y)=\chi_{R}(x)(\varphi(x)-\varphi(y))+\varphi(y)(\chi_{R}(x)-\chi_{R}(y))$ and $|\varphi(y)|\le|\varphi(0)|+2RL$, so, using $R\ge1$,

(2b) The identity for $f_{R}$. For $k\in\mathbb{N}$ with $k\ge1$ let $g_{R,k}=f_{R}*\rho_{1/k}$. Since $f_{R}$ vanishes off $[-2R,2R]$, (0g) shows that $g_{R,k}\in C_{c}^{\infty}(\mathbb{R})$, that $g_{R,k}$ is Lipschitz with constant $K$, and that $|g_{R,k}-f_{R}|\le K/k$. By Step 1, $\int\xi g_{R,k}\,d\mu=\int Q_{g_{R,k}}\,d(\mu\boxtimes\mu)$. As $k\to\infty$: $g_{R,k}(x)\to f_{R}(x)$ for every $x$, and $|\xi g_{R,k}|\le|\xi|(|\varphi|+K)$, which is integrable by (0b); so $\int\xi g_{R,k}\,d\mu\to\int\xi f_{R}\,d\mu$ by dominated convergence. Also $Q_{g_{R,k}}(x,y)\to Q_{f_{R}}(x,y)$ for $x\ne y$, both vanish on $\Delta$, and $|Q_{g_{R,k}}|\le K$ by (0d); so $\int Q_{g_{R,k}}\,d(\mu\boxtimes\mu)\to\int Q_{f_{R}}\,d(\mu\boxtimes\mu)$ by dominated convergence. Hence $\int\xi f_{R}\,d\mu=\int Q_{f_{R}}\,d(\mu\boxtimes\mu)$.

(2c) Removing the cutoff. Let $R=j\in\mathbb{N}$, $j\ge1$, and let $j\to\infty$. For every $x$, $f_{j}(x)=\varphi(x)$ once $j\ge|x|$, and $|\xi f_{j}|\le|\xi\varphi|$; so $\int\xi f_{j}\,d\mu\to\int\xi\varphi\,d\mu$ by dominated convergence. For every $(x,y)$, $Q_{f_{j}}(x,y)=Q_{\varphi}(x,y)$ once $j\ge\max(|x|,|y|)$, and $|Q_{f_{j}}|\le K$ by (2a) and (0d); so $\int Q_{f_{j}}\,d(\mu\boxtimes\mu)\to\int Q_{\varphi}\,d(\mu\boxtimes\mu)$. With (2b) this proves (3).

Step 3 ($\langle\Xi_{\mu},\mathrm{id}\rangle_{\mu}=1$). The identity map is Lipschitz with constant $1$, and $Q_{\mathrm{id}}(x,y)=1$ for $x\ne y$, $Q_{\mathrm{id}}=0$ on $\Delta$. By (3) and (0a), $\int\xi(x)\,x\,\mu(dx)=(\mu\boxtimes\mu)(\mathbb{R}^{2}\setminus\Delta)=1$. The class of $\mathrm{id}$ lies in $L^{2}(\mu;\mathbb{R})$ by Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §identity, so by (0b) $\langle\Xi_{\mu},\mathrm{id}\rangle_{\mu}=1$.

Step 4 (The optimal coupling is induced by a monotone map). Since $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R})$ and $\mu$ is atomless, On the Real Line an Atomless Source is Uniquely Mapped, by a Nondecreasing Optimal Map §monotone gives $E\in\mathcal{B}(\mathbb{R})$ with $\mu(E)=1$ and a Borel map $S:\mathbb{R}\to\mathbb{R}$ with $S(x)\le S(x')$ for all $x,x'\in E$ with $x\le x'$ and $S=0$ off $E$, which is an optimal map from $\mu$ to $\nu$: $S_{\#}\mu=\nu$ and $(\mathrm{id},S)_{\#}\mu$ is an optimal coupling. By On the Real Line an Atomless Source is Uniquely Mapped, by a Nondecreasing Optimal Map §uniquely-mapped and Optimal Transport Maps and Uniquely Mapped Pairs of Probability Measures §uniquely-mapped there is an optimal map $T_{0}$ such that every optimal coupling of $\mu$ and $\nu$ equals $(\mathrm{id},T_{0})_{\#}\mu$; applied to $(\mathrm{id},S)_{\#}\mu$ and to $\pi$ this gives $\pi=(\mathrm{id},T_{0})_{\#}\mu=(\mathrm{id},S)_{\#}\mu$. By The Optimal Map as a Square-Integrable Vector Field: Integrability, Transport Cost and Uniqueness of the Class §square-integrable, $\int S^{2}\,d\mu=M_{2}(\nu)<\infty$, the class of $S$ lies in $L^{2}(\mu;\mathbb{R})$, and so does $\mathrm{id}-S$; in particular $\int(S(x)-x)^{2}\,\mu(dx)<\infty$ (One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §scalars). The displacement pairing of the statement is that of The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §pairing (The Intrinsic Calculus on the Wasserstein Space: Standing Notation §couplings), so The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §displacement with $d=1$ and $\eta=-\Xi_{\mu}$ yields, using (0b), claim 2 of Linearity and Monotonicity of the Lebesgue Integral and Step 3,

Step 5 (Transporting the energy of $\nu$). Let $\hat S:\mathbb{R}^{2}\to\mathbb{R}^{2}$, $\hat S(x,y)=(S(x),S(y))$, the pairing of $S\circ\mathrm{pr}_{1}$ and $S\circ\mathrm{pr}_{2}$, Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing. For every Borel $F:\mathbb{R}^{2}\to[0,\infty]$,

By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product, the left side is $\int_{\mathbb{R}\times\mathbb{R}}F\circ\iota\,d(\nu\otimes\nu)$, with $F\circ\iota$ measurable for the product $\sigma$-algebra. By the Sections and Tonelli clauses of Tonelli and Fubini Theorems, each section $y'\mapsto F(x',y')$ is Borel, the map $G(x')=\int F(x',y')\,\nu(dy')$ is Borel with values in $[0,\infty]$, and the left side equals $\int G\,d\nu$. By the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward (with $\nu=S_{\#}\mu$), $G(x')=\int F(x',S(y))\,\mu(dy)$ for every $x'$, and $\int G\,d\nu=\int G(S(x))\,\mu(dx)$. On the other hand $F\circ\hat S$ is Borel, $(F\circ\hat S)(x,y)=F(S(x),S(y))$, and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product together with the same two clauses of Tonelli and Fubini Theorems, now for $\mu\otimes\mu$, give $\int F\circ\hat S\,d(\mu\boxtimes\mu)=\int\bigl(\int F(S(x),S(y))\,\mu(dy)\bigr)\mu(dx)=\int G(S(x))\,\mu(dx)$. This proves (5).

(5a) With $F=\mathbf{1}_{\Delta}$ (Borel by claim 1 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions): the Borel set $\hat S^{-1}(\Delta)=\{(x,y):S(x)=S(y)\}$ has $(\mu\boxtimes\mu)(\hat S^{-1}(\Delta))=(\nu\boxtimes\nu)(\Delta)=0$ by (0a).

(5b) Let $\ell^{+}=\max(\ell,0)$ and $\ell^{-}=\max(-\ell,0)$, Borel by claim 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. As $\nu\in\mathcal{D}_{\log}$, $\ell$ is $\nu\boxtimes\nu$-integrable, so $\int\ell^{\pm}\,d(\nu\boxtimes\nu)<\infty$, and by (5) $\int\ell^{\pm}\circ\hat S\,d(\mu\boxtimes\mu)=\int\ell^{\pm}\,d(\nu\boxtimes\nu)$. Hence $\ell\circ\hat S=\ell^{+}\circ\hat S-\ell^{-}\circ\hat S$ is $\mu\boxtimes\mu$-integrable and, by Integrable Function and the Lebesgue Integral and The Logarithmic Energy of a Probability Measure on the Real Line §energy,

Step 6 (The monotone quotient of $S$ is controlled by the score). Let $R=Q_{S}\,\mathbf{1}_{E\times E}$, Borel by (0c) and claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. For $x,y\in E$ with $x\ne y$, $S(x)-S(y)$ and $x-y$ are both $\ge0$ or both $\le0$ by monotonicity of $S$ on $E$, so $R\ge0$ everywhere. We show $\int R\,d(\mu\boxtimes\mu)\le\int\xi S\,d\mu$; in particular $R$ is integrable.

For $n\in\mathbb{N}$ with $n\ge1$ let $T_{n}(t)=\max(-n,\min(t,n))$. Checking the cases $t<-n$, $|t|\le n$, $t>n$: $T_{n}$ is nondecreasing, Lipschitz with constant $1$ (so Borel), $|T_{n}(t)|\le\min(|t|,n)$, and $T_{n}(t)=t$ when $|t|\le n$. Define $h_{n}:\mathbb{R}\to\mathbb{R}$ by

the supremum of a nonempty set bounded above by $n$. Then (h1) $-n\le h_{n}\le n$; (h2) $h_{n}$ is nondecreasing on $\mathbb{R}$, the set grows with $x$; (h3) $h_{n}(x)=T_{n}(S(x))$ for $x\in E$, because $T_{n}\circ S$ is nondecreasing on $E$, so $T_{n}(S(x))\ge-n$ is the largest element of the set. For $x\in\mathbb{R}$ put $h_{n}(x^{-})=\sup\{h_{n}(y):y<x\}$, so $-n\le h_{n}(x^{-})\le h_{n}(x)$, and let $D_{n}=\{x:h_{n}(x^{-})<h_{n}(x)\}$.

(6a) $D_{n}$ is Borel and $\mu(D_{n})=0$. For $k\in\mathbb{N}$ let $D_{n,k}=\{x:h_{n}(x)-h_{n}(x^{-})\ge1/(k+1)\}$, so $D_{n}=\bigcup_{k}D_{n,k}$. If $x_{1}<\dots<x_{N}$ are points of $D_{n,k}$, then $h_{n}(x_{1})\ge-n+1/(k+1)$, and for $2\le i\le N$ we have $h_{n}(x_{i}^{-})\ge h_{n}(x_{i-1})$, hence $h_{n}(x_{i})\ge h_{n}(x_{i-1})+1/(k+1)$; so $n\ge h_{n}(x_{N})\ge-n+N/(k+1)$ and $N\le2n(k+1)$. Thus $D_{n,k}$ has at most $2n(k+1)$ elements; it is finite, hence countable by claim 2 of Basic Properties of Countable Sets, and $D_{n}$ is countable by A Countable Union of Countable Sets is Countable. Since $\mu$ is atomless, Countable Sets are Null for an Atomless Measure, One-Point Sets are Lebesgue Null, and an Absolutely Continuous Measure is Atomless §countable gives $D_{n}\in\mathcal{B}(\mathbb{R})$ and $\mu(D_{n})=0$.

(6b) Lipschitz approximants. For $m\in\mathbb{N}$ with $m\ge1$ let

a real number since every element of the set is $\ge-n$. Then $-n\le g_{n,m}\le h_{n}\le n$ (take $y=x$). $g_{n,m}$ is Lipschitz with constant $m$: for all $x,x',y$, $g_{n,m}(x)\le h_{n}(y)+m|x'-y|+m|x-x'|$, and taking the infimum over $y$ gives $g_{n,m}(x)\le g_{n,m}(x')+m|x-x'|$; exchange $x,x'$. $g_{n,m}$ is nondecreasing: if $x\le x'$ and $a=x'-x$, then for every $y$, by (h2), $g_{n,m}(x)\le h_{n}(y-a)+m|x-(y-a)|\le h_{n}(y)+m|x'-y|$, and taking the infimum over $y$ gives $g_{n,m}(x)\le g_{n,m}(x')$. Consequently $Q_{g_{n,m}}\ge0$ everywhere.

(6c) Convergence off $D_{n}$. Let $x\notin D_{n}$ and $\eta>0$. As $h_{n}(x^{-})=h_{n}(x)$ there is $y_{0}<x$ with $h_{n}(y_{0})>h_{n}(x)-\eta$. Let $m\ge2n/(x-y_{0})$. For $y\ge y_{0}$, $h_{n}(y)+m|x-y|\ge h_{n}(y_{0})>h_{n}(x)-\eta$ by (h2); for $y<y_{0}$, $h_{n}(y)+m|x-y|\ge-n+m(x-y_{0})\ge n\ge h_{n}(x)$ by (h1). So $h_{n}(x)-\eta\le g_{n,m}(x)\le h_{n}(x)$. Hence $g_{n,m}(x)\to h_{n}(x)$ as $m\to\infty$.

(6d) Fatou in $m$. Let $R_{n}=Q_{T_{n}\circ S}\,\mathbf{1}_{E\times E}$, Borel as $R$ is and $\ge0$ as $T_{n}\circ S$ is nondecreasing on $E$. Let $G_{n}=((E\setminus D_{n})\times(E\setminus D_{n}))\setminus\Delta$; its complement lies in $((\mathbb{R}\setminus(E\setminus D_{n}))\times\mathbb{R})\cup(\mathbb{R}\times(\mathbb{R}\setminus(E\setminus D_{n})))\cup\Delta$, which is $\mu\boxtimes\mu$-null by (0a) and (6a). For $(x,y)\in G_{n}$, (6c) and (h3) give $Q_{g_{n,m}}(x,y)\to Q_{h_{n}}(x,y)=Q_{T_{n}\circ S}(x,y)=R_{n}(x,y)$; at a point where a sequence of nonnegative reals converges, its lower limit in the sense of Fatou's Lemma is its limit. So $\liminf_{m}Q_{g_{n,m}}=R_{n}$ almost everywhere, and by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison, Fatou, and (3) applied to the Lipschitz functions $g_{n,m}$,

By (6c), (h3) and (6a), $g_{n,m}\to T_{n}\circ S$ on the $\mu$-full set $E\setminus D_{n}$, and $|\xi g_{n,m}|\le n|\xi|$; by dominated convergence $\int\xi g_{n,m}\,d\mu\to\int\xi\,(T_{n}\circ S)\,d\mu$. Hence $\int R_{n}\,d(\mu\boxtimes\mu)\le\int\xi\,(T_{n}\circ S)\,d\mu$.

(6e) Fatou in $n$. For every $(x,y)$, $R_{n}(x,y)\to R(x,y)$: if $x,y\in E$ and $x\ne y$ then $R_{n}(x,y)=R(x,y)$ once $n\ge\max(|S(x)|,|S(y)|)$, and otherwise both vanish. By Fatou and (6d),

the last by dominated convergence: $T_{n}(S(x))\to S(x)$ for every $x$, and $|\xi\,(T_{n}\circ S)|\le|\xi S|$, integrable by (0b) since $\int S^{2}\,d\mu<\infty$.

Step 7 (Conclusion). First, $\log t\le t-1$ for every real $t>0$. Indeed $\exp(u)\ge1+u$ for all $u\in\mathbb{R}$: for $u\ge0$ by claim 4 of Basic Properties of the Exponential Function; for $u<0$, the function $f(v)=\exp(v)-1-v$ is differentiable at every point with $f'(v)=\exp(v)-1$ (claim 3 of Basic Properties of the Exponential Function and claims 1 and 2 of Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives), hence continuous by (0-), so Mean Value Theorem on a Closed Real Interval, applied to the restriction of $f$ to $[u,0]$, gives $c\in(u,0)$ with $f(0)-f(u)=(\exp(c)-1)(-u)<0$, as $\exp(c)<\exp(0)=1$ by claims 1 and 4 of Basic Properties of the Exponential Function; thus $f(u)>f(0)=0$. With $u=\log t$ and $\exp(\log t)=t$ (The Natural Logarithm) this is $t\ge1+\log t$.

Let $G=\{(x,y)\in E\times E:x\ne y,\ S(x)\ne S(y)\}=(E\times E)\setminus(\Delta\cup\hat S^{-1}(\Delta))$, Borel. Its complement lies in $((\mathbb{R}\setminus E)\times\mathbb{R})\cup(\mathbb{R}\times(\mathbb{R}\setminus E))\cup\Delta\cup\hat S^{-1}(\Delta)$, which is $\mu\boxtimes\mu$-null by (0a) and (5a). Let $(x,y)\in G$. Then $R(x,y)=Q_{S}(x,y)$ is nonzero and $\ge0$, hence $>0$, and $|S(x)-S(y)|=R(x,y)\,|x-y|$. By the definition of $\ell$ in The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure and $\log(st)=\log s+\log t$ (The Natural Logarithm),