Proof of Basic Properties of the Logarithmic Energy on the Real Line: Lower Bound, Translation Invariance, Closed Sublevel Sets and Lower Semicontinuity

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 for adding, multiplying and comparing inequalities between real numbers in Elementary Order Arithmetic in an Ordered Field and Elementary Arithmetic in an Ordered Field, and Properties of the Absolute Value in an Ordered Field, are used without further mention. As in One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative, a point of $\mathbb{R}^{2}$ is written $(x,y)=\iota(x,y)$, and $\lVert s\rVert=|s|$, $\lVert s\rVert^{2}=s^{2}$ for $s\in\mathbb{R}$ by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative (i); in particular $M_{2}(\nu)=\int_{\mathbb{R}}x^{2}\,\nu(dx)$ for $\nu\in\mathcal{P}(\mathbb{R})$ (The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (i)). The projections $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{2}\to\mathbb{R}$ satisfy $\mathrm{pr}_{1}(x,y)=x$, $\mathrm{pr}_{2}(x,y)=y$ and are Borel (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i)); $\ell$ is the logarithmic kernel, Borel by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure (i), and $\Delta$ the diagonal of that lemma, a Borel set by the same clause. Two consequences of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iii) and the Tonelli part of Tonelli's and Fubini's Theorems are used throughout: for $\nu,\nu'\in\mathcal{P}(\mathbb{R})$ and Borel $F:\mathbb{R}^{2}\to[0,\infty]$, each section $y\mapsto F(x,y)$ is Borel, $x\mapsto\int F(x,y)\,\nu(dy)$ is Borel, and

$$\int_{\mathbb{R}^{2}}F\,d(\nu'\boxtimes\nu)=\int_{\mathbb{R}}\Bigl(\int_{\mathbb{R}}F(x,y)\,\nu(dy)\Bigr)\nu'(dx),\tag{T}$$

also with the order of integration reversed; and, the images of $\nu\boxtimes\nu$ under $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ being $\nu$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iii)), $\int u\circ\mathrm{pr}_{i}\,d(\nu\boxtimes\nu)=\int u\,d\nu$ for $i=1,2$ and every Borel $u:\mathbb{R}\to[0,\infty]$, by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv).

Step 0 (A quadratic minorant of the kernel). Define $q:\mathbb{R}^{2}\to\mathbb{R}$ by $q(z)=1+\tfrac12\bigl(\mathrm{pr}_{1}(z)^{2}+\mathrm{pr}_{2}(z)^{2}\bigr)$; it is Borel by claims 1 to 3 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps and continuous by Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit (i) and claims 1, 2, 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space. For real $s$, $0\le(|s|-1)^{2}=s^{2}-2|s|+1$ by Nonnegativity of Squares in an Ordered Field, so $|s|\le\tfrac12(1+s^{2})$; hence $|x|+|y|\le q(x,y)$ and, by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure (ii),

$$-q(z)\le\ell(z)\qquad(z\in\mathbb{R}^{2}).\tag{Q}$$

For $\nu\in\mathcal{P}_{2}(\mathbb{R})$, the facts above and claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion give $\int q\,d(\nu\boxtimes\nu)=1+M_{2}(\nu)<\infty$, so $q$ is $\nu\boxtimes\nu$-integrable (Integrable Functions and Their Lebesgue Integral).

Clause 1. Let $\mu\in\mathcal{D}_{\log}$. Both $\ell$ and $-q$ are $\mu\boxtimes\mu$-integrable, so (Q) and claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion give $\mathcal{E}_{\log}(\mu)=\int\ell\,d(\mu\boxtimes\mu)\ge-\int q\,d(\mu\boxtimes\mu)=-(1+M_{2}(\mu))$.

Clause 2. Let $\mu\in\mathcal{D}_{\log}$, $a\in\mathbb{R}$ and $\nu=(\tau_{a})_{\#}\mu$, where $\tau_{a}(x)=x+a$ (The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation (vi)). Since $|\tau_{a}(x)-\tau_{a}(x')|=|x-x'|$, $\tau_{a}$ is Lipschitz, hence continuous (A Lipschitz Map is Uniformly Continuous and Continuous) and Borel (claims 3(a) and 5 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 $\nu\in\mathcal{P}(\mathbb{R})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv). By the change of variables there and the inequality $\lVert x-y\rVert^{2}\le2\lVert x\rVert^{2}+2\lVert y\rVert^{2}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv) with $y=-a$, $M_{2}(\nu)=\int(x+a)^{2}\,\mu(dx)\le2M_{2}(\mu)+2a^{2}<\infty$. For $b\in\mathbb{R}$, $\nu(\{b\})=\mu(\tau_{a}^{-1}(\{b\}))=\mu(\{b-a\})=0$. Now let $F:\mathbb{R}^{2}\to[0,\infty]$ be Borel with $F(x+a,y+a)=F(x,y)$ for all $x,y$. By (T), the change of variables of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) applied first to the inner integrand $y\mapsto F(x,y)$ and then to the Borel function $x\mapsto\int F(x,y)\,\nu(dy)$, and (T) again,

$$\int F\,d(\nu\boxtimes\nu)=\int\Bigl(\int F(x,y+a)\,\mu(dy)\Bigr)\nu(dx)=\int\Bigl(\int F(x+a,y+a)\,\mu(dy)\Bigr)\mu(dx)=\int F\,d(\mu\boxtimes\mu).$$

Since $(x+a)-(y+a)=x-y$, $\ell(x+a,y+a)=\ell(x,y)$ for all $x,y$, so this applies to the positive and negative parts $\ell^{+},\ell^{-}$ of $\ell$, which are Borel (Integrable Functions and Their Lebesgue Integral). Both integrals are finite for $\mu\boxtimes\mu$ because $\ell$ is $\mu\boxtimes\mu$-integrable; hence $\ell$ is $\nu\boxtimes\nu$-integrable and $\int\ell\,d(\nu\boxtimes\nu)=\int\ell^{+}d(\nu\boxtimes\nu)-\int\ell^{-}d(\nu\boxtimes\nu)=\mathcal{E}_{\log}(\mu)$. By The Logarithmic Energy of a Probability Measure on the Real Line (i), $\nu\in\mathcal{D}_{\log}$ and $\mathcal{E}_{\log}(\nu)=\mathcal{E}_{\log}(\mu)$.

Clause 3. Let $(\mu_{n})_{n}$, $\mu$ and $c$ be as in clause 3.

Step 1 (Truncated kernels). Write $\rho(z)=|\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)|$, so $\Delta=\{\rho=0\}$. For $k\in\mathbb{N}$ put $s_{k}=\exp(-k)>0$ and $\ell_{k}(z)=-\log\bigl(\max\{\rho(z),s_{k}\}\bigr)$, where $\exp$ and $\log$ are as in The Natural Logarithm. From that definition and claims 1, 2 and 4 of Basic Properties of the Exponential Function: $\log(\exp u)=u$, $\log(st)=\log s+\log t$, $1/s_{k}=\exp(k)$; $\log$ is nondecreasing and $\log s<\log t$ for $0<s<t$ (else $t=\exp(\log t)\le\exp(\log s)=s$); and for $s_{k}\le t\le s$, with $v=\log(s/t)\ge0$, $s/t=\exp(v)\ge1+v$, whence $0\le\log s-\log t=v\le(s-t)/t\le\exp(k)(s-t)$.

(i) For $z,z'\in\mathbb{R}^{2}$ the last estimate, applied to the larger and the smaller of $\max\{\rho(z),s_{k}\}$ and $\max\{\rho(z'),s_{k}\}$, together with $|\max\{\rho(z),s_{k}\}-\max\{\rho(z'),s_{k}\}|\le|\rho(z)-\rho(z')|$ (a case check using claim 2 of Elementary Properties of the Maximum of Two Elements) and $|\rho(z)-\rho(z')|\le|\mathrm{pr}_{1}(z)-\mathrm{pr}_{1}(z')|+|\mathrm{pr}_{2}(z)-\mathrm{pr}_{2}(z')|\le2\lVert z-z'\rVert$ (claims 7, 5 of Properties of the Absolute Value in an Ordered Field and Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit (i)), gives $|\ell_{k}(z)-\ell_{k}(z')|\le2\exp(k)\lVert z-z'\rVert$. So $\ell_{k}$ is continuous (A Lipschitz Map is Uniformly Continuous and Continuous) and Borel (claims 3(a), 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets).

(ii) On $\Delta$, $\ell_{k}=-\log s_{k}=k$. Off $\Delta$, $\rho(z)>0$ and $\ell(z)=-\log\rho(z)$; if $\rho(z)\ge s_{k}$ then $\ell_{k}(z)=\ell(z)\le k$, and if $\rho(z)<s_{k}$ then $\ell_{k}(z)=k<\ell(z)$; so $\ell_{k}=\min\{\ell,k\}$ off $\Delta$.

(iii) Put $g_{k}=\ell_{k}+q$, a continuous Borel function. By (Q) and $0\le k$, $g_{k}\ge0$ everywhere: off $\Delta$ both $\ell$ and $k$ are at least $-q$, and on $\Delta$, $g_{k}=k+q\ge0$. Also $g_{k}\le\ell+q$ off $\Delta$, and $\ell+q\ge0$ everywhere by (Q).

Step 2 (Weak convergence of the product measures). We show that $(\mu_{n}\boxtimes\mu_{n})_{n}$ converges weakly to $\mu\boxtimes\mu$ on $(\mathbb{R}^{2},d_{E})$. For each $n$ let $\pi_{n}\in\Pi(\mu_{n},\mu)$ be an optimal coupling, $I(\pi_{n})=W_{2}(\mu_{n},\mu)^{2}$, given by Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment. Let $F:\mathbb{R}^{2}\to\mathbb{R}$ be bounded, with bound $B$, and Lipschitz with constant $\Lambda$; it is continuous, hence Borel, as in Step 1(i). By claim 4 of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space and claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, $d_{E}((x,y),(x,y'))=|y-y'|$ and $d_{E}((x,y),(x',y))=|x-x'|$. Hence for each $x$ the function $F^{x}(y)=F(x,y)+B$ is bounded, nonnegative and Lipschitz with constant $\Lambda$, and by 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 (viii) (with $d=1$ and $\pi_{n}$)

$$\Bigl|\int F^{x}\,d\mu_{n}-\int F^{x}\,d\mu\Bigr|\le\Lambda\sqrt{I(\pi_{n})}=\Lambda\,W_{2}(\mu_{n},\mu).$$

Put $\Psi_{n}(x)=\int F^{x}\,d\mu_{n}$ and $\Psi(x)=\int F^{x}\,d\mu$; they are Borel by (T), with values in $[0,2B]$, and $|\Psi(x)-\Psi(x')|\le\int|F(x,y)-F(x',y)|\,\mu(dy)\le\Lambda|x-x'|$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, so $\Psi$ is bounded and Lipschitz with constant $\Lambda$. By (T) applied to $F+B\ge0$ and claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion,

$$\Bigl|\int(F+B)\,d(\mu_{n}\boxtimes\mu_{n})-\int(F+B)\,d(\mu\boxtimes\mu)\Bigr|\le\int|\Psi_{n}-\Psi|\,d\mu_{n}+\Bigl|\int\Psi\,d\mu_{n}-\int\Psi\,d\mu\Bigr|\le2\Lambda\,W_{2}(\mu_{n},\mu),$$

the second term by 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 (viii) applied to $\Psi$. Since $\int(F+B)\,d\lambda=\int F\,d\lambda+B$ for every probability measure $\lambda$ on $\mathbb{R}^{2}$, and $W_{2}(\mu_{n},\mu)\to0$, claim 3 of Order Properties of Limits of Real Sequences gives $\int F\,d(\mu_{n}\boxtimes\mu_{n})\to\int F\,d(\mu\boxtimes\mu)$. As $F$ was an arbitrary bounded Lipschitz function, claim 1 of Portmanteau Theorem on a Metric Space gives the weak convergence.

Step 3 (A uniform bound). Put $C_{1}=c+1+M_{2}(\mu)$. By The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance (ii), $|\sqrt{M_{2}(\mu_{n})}-\sqrt{M_{2}(\mu)}|\le W_{2}(\mu_{n},\mu)$, so $\sqrt{M_{2}(\mu_{n})}\to\sqrt{M_{2}(\mu)}$ by claim 3 of Order Properties of Limits of Real Sequences and $M_{2}(\mu_{n})\to M_{2}(\mu)$ by claim 2 of Arithmetic of Limits of Real Sequences. Let $k,j\in\mathbb{N}$ and $f_{k,j}=\min\{g_{k},j\}$, which is continuous (claim 4 of Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space) with values in $[0,j]$. For each $n$, $\mu_{n}\in\mathcal{D}_{\log}$ has no atoms, so $(\mu_{n}\boxtimes\mu_{n})(\Delta)=0$ by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure (iii); by Step 1(iii), $f_{k,j}\le\ell+q$ off $\Delta$, so The Lebesgue Integral and Null Sets (iii) gives $\int f_{k,j}\,d(\mu_{n}\boxtimes\mu_{n})\le\int(\ell+q)\,d(\mu_{n}\boxtimes\mu_{n})$, where the right side, the integral of a nonnegative integrable function, equals $\mathcal{E}_{\log}(\mu_{n})+1+M_{2}(\mu_{n})\le c+1+M_{2}(\mu_{n})$ by Step 0 and claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion. By Step 2 and the definition of weak convergence, claim 1 of Order Properties of Limits of Real Sequences and claim 1 of Arithmetic of Limits of Real Sequences, $\int f_{k,j}\,d(\mu\boxtimes\mu)\le C_{1}$. For fixed $k$, $(f_{k,j})_{j}$ is nondecreasing with supremum $g_{k}$ (claim 1 of The Archimedean Property of the Real Numbers), so Monotone Convergence Theorem gives

$$\int g_{k}\,d(\mu\boxtimes\mu)\le C_{1}\qquad(k\in\mathbb{N}).\tag{B}$$

Step 4 ($\mu$ has no atoms). Suppose $p=\mu(\{a\})>0$ for some $a\in\mathbb{R}$. The set $S=\iota(\{a\}\times\{a\})$ is Borel with $(\mu\boxtimes\mu)(S)=p^{2}>0$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iii), Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (v)), and $S=\{(a,a)\}\subseteq\Delta$, where $g_{k}(a,a)=k+q(a,a)\ge k$ by Step 1(ii). So $g_{k}\ge k\mathbf{1}_{S}$ and (B) gives $kp^{2}\le C_{1}$ for every $k\in\mathbb{N}$, contradicting claim 2 of The Archimedean Property of the Real Numbers. Hence $\mu(\{a\})=0$ for every $a$, and $(\mu\boxtimes\mu)(\Delta)=0$ by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure (iii).

Step 5 ($\mu\in\mathcal{D}_{\log}$ and $\mathcal{E}_{\log}(\mu)\le c$). Let $u_{k}=g_{k}\mathbf{1}_{\mathbb{R}^{2}\setminus\Delta}$. By Step 1(ii), off $\Delta$, $u_{k}=\min\{\ell,k\}+q$, which is nondecreasing in $k$ and equals $\ell+q$ once $k>\ell$ (claim 1 of The Archimedean Property of the Real Numbers); on $\Delta$, $u_{k}=0$. So $(u_{k})_{k}$ is nondecreasing with supremum $v=(\ell+q)\mathbf{1}_{\mathbb{R}^{2}\setminus\Delta}$, and Monotone Convergence Theorem, $u_{k}\le g_{k}$ and (B) give $\int v\,d(\mu\boxtimes\mu)\le C_{1}$. As $\ell+q=v$ off the null set $\Delta$ and both are nonnegative, The Lebesgue Integral and Null Sets (iii) gives $\int(\ell+q)\,d(\mu\boxtimes\mu)\le C_{1}<\infty$, so $\ell+q$ is $\mu\boxtimes\mu$-integrable (Integrable Functions and Their Lebesgue Integral). By Step 0 and claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, $\ell=(\ell+q)-q$ is integrable with $\int\ell\,d(\mu\boxtimes\mu)\le C_{1}-(1+M_{2}(\mu))=c$. Since $\mu\in\mathcal{P}_{2}(\mathbb{R})$ and $\mu$ has no atoms (Step 4), The Logarithmic Energy of a Probability Measure on the Real Line (i) gives $\mu\in\mathcal{D}_{\log}$ and $\mathcal{E}_{\log}(\mu)\le c$.

Clause 4. Let $\mu\in\mathcal{D}_{\log}$ and $\varepsilon>0$, and suppose that no $\delta>0$ has the property required in Lower Semicontinuous Function on a Subset of a Metric Space. Then for every $j\in\mathbb{N}$ the set $A_{j}$ of those $\nu\in\mathcal{D}_{\log}$ with $W_{2}(\mu,\nu)<1/(j+1)$ and $\mathcal{E}_{\log}(\nu)\le\mathcal{E}_{\log}(\mu)-\varepsilon$ is nonempty, the order of $\mathbb{R}$ being total. By Axiom of Countable Choice there are $\mu_{j}\in A_{j}$ for all $j$. Given $\varepsilon'>0$, claim 2 of The Archimedean Property of the Real Numbers gives $N$ with $1<N\varepsilon'$, so $W_{2}(\mu_{j},\mu)=W_{2}(\mu,\mu_{j})<1/(j+1)<\varepsilon'$ for $j\ge N$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space (i)); thus $(W_{2}(\mu_{j},\mu))_{j}$ has limit $0$. Clause 3 with $c=\mathcal{E}_{\log}(\mu)-\varepsilon$ gives $\mathcal{E}_{\log}(\mu)\le\mathcal{E}_{\log}(\mu)-\varepsilon$, which contradicts $\varepsilon>0$. Hence $\mathcal{E}_{\log}$ is lower semicontinuous at every $\mu\in\mathcal{D}_{\log}$ relative to $\mathcal{D}_{\log}$, that is, lower semicontinuous on $\mathcal{D}_{\log}$.