Proof of The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure

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 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 $(x,y)$, meaning $\iota(x,y)$; by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections the coordinate projections $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{2}\to\mathbb{R}$ satisfy $\mathrm{pr}_{1}(x,y)=x$ and $\mathrm{pr}_{2}(x,y)=y$, and every $z\in\mathbb{R}^{2}$ equals $(\mathrm{pr}_{1}(z),\mathrm{pr}_{2}(z))$. Define $\rho:\mathbb{R}^{2}\to\mathbb{R}$ by $\rho(z)=|\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)|$, so that $\rho(x,y)=|x-y|$, and write $\exp$ for the exponential function of The Natural Logarithm.

Step 1 (Three properties of the logarithm). By The Natural Logarithm, $\log(\exp(u))=u$ and $\exp(\log t)=t$ for real $u$ and $t>0$, and $\log(st)=\log s+\log t$ for $s,t>0$. By claim 1 of Basic Properties of the Exponential Function, $\exp(0)=1$, hence $\log 1=0$; and $\log(\exp(-u))=-u$ for every real $u$.

(a) Monotonicity. Let $0<s<t$. If $\log t\le\log s$, then, $\exp$ being strictly increasing by claim 4 of Basic Properties of the Exponential Function, $t=\exp(\log t)\le\exp(\log s)=s$, which is false; so $\log s<\log t$. Consequently $\log s\le\log t$ whenever $0<s\le t$; in particular $0\le\log t$ for $t\ge1$ and $\log t<0$ for $0<t<1$.

(b) A Lipschitz bound. Let $s_{0},s,t$ be real with $0<s_{0}\le t\le s$, and put $v=\log(s/t)$. Then $\log s=\log t+v$, and $0\le v$ by (a) since $1\le s/t$. By claim 4 of Basic Properties of the Exponential Function, $\exp(v)\ge1+v$ because $v\ge0$, that is $s/t\ge1+v$. Hence

(c) $\log t\le t$ for every $t>0$. If $t\ge1$, then $v=\log t\ge0$ by (a), and $t=\exp(v)\ge1+v>v$ by claim 4 of Basic Properties of the Exponential Function. If $0<t<1$, then $\log t<0<t$ by (a).

Step 2 (The diagonal is Borel). The map $z\mapsto\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert$ on $\mathbb{R}^{2}=\mathbb{R}^{1+1}$ is Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, and it equals $\rho$ because $\lVert s\rVert=|s|$ for $s\in\mathbb{R}=\mathbb{R}^{1}$ by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §scalars; so $\rho$ is Borel. For $z\in\mathbb{R}^{2}$, $z\in\Delta$ holds if and only if $\mathrm{pr}_{1}(z)=\mathrm{pr}_{2}(z)$, that is, by claim 1 of Properties of the Absolute Value in an Ordered Field, if and only if $\rho(z)=0$. Thus $\Delta=\rho^{-1}(\{0\})$, and since $\{0\}\in\mathcal{B}(\mathbb{R})$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §finite-sets, $\Delta\in\mathcal{B}(\mathbb{R}^{2})$ by the definition of a Borel map. So is $\mathbb{R}^{2}\setminus\Delta$.

Step 3 (Truncated kernels). For $k\in\mathbb{N}$ let $s_{k}=\exp(-k)$, which is positive with $1/s_{k}=\exp(k)$ by claim 2 of Basic Properties of the Exponential Function, and define $\ell_{k}:\mathbb{R}^{2}\to\mathbb{R}$ by $\ell_{k}(z)=-\log\bigl(\max\{\rho(z),s_{k}\}\bigr)$, which is defined because $\max\{\rho(z),s_{k}\}\ge s_{k}>0$ by claim 1 of Elementary Properties of the Maximum of Two Elements.

(i) $\ell_{k}$ is Borel. Let $z,z'\in\mathbb{R}^{2}$ and put $s=\max\{\rho(z),s_{k}\}$, $t=\max\{\rho(z'),s_{k}\}$. Both are at least $s_{k}$, so Step 1(b), applied with the larger of $s,t$ in the role of $s$ there, gives $|\log s-\log t|\le\exp(k)\,|s-t|$. Moreover $|s-t|\le|\rho(z)-\rho(z')|$: by claim 2 of Elementary Properties of the Maximum of Two Elements each maximum is attained; if both equal $s_{k}$ then $s-t=0$; if $s=\rho(z)\ge s_{k}$ and $t=s_{k}\ge\rho(z')$ then $0\le s-t\le\rho(z)-\rho(z')$; the case with the roles exchanged is symmetric; and if $s=\rho(z)$, $t=\rho(z')$ there is equality. By claims 7, 5 and 2 of Properties of the Absolute Value in an Ordered Field,

the last step because both projections are Lipschitz with constant $1$ by Continuity of the Coordinate Projections and of the Quadratic Cost Function, and Passage of a Cost Bound to a Weak Limit §projections, the Euclidean distance of $\mathbb{R}^{1}$ being the absolute-value metric as recorded in One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative. Hence $|\ell_{k}(z)-\ell_{k}(z')|\le2\exp(k)\lVert z-z'\rVert$, so $\ell_{k}$ is Lipschitz, therefore continuous by A Lipschitz Map is Uniformly Continuous, therefore Borel by 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.

(ii) Values of $\ell_{k}$. If $z\in\Delta$, then $\rho(z)=0$ and $\ell_{k}(z)=-\log s_{k}=k$. If $z\notin\Delta$, then $\rho(z)>0$ and $\ell(z)=-\log\rho(z)$; if $\rho(z)\ge s_{k}$ then $\ell_{k}(z)=\ell(z)$ and $\ell(z)\le k$ because $\log\rho(z)\ge\log s_{k}=-k$ by Step 1(a); if $\rho(z)<s_{k}$ then $\ell_{k}(z)=k<\ell(z)$ by Step 1(a). Hence $\ell_{k}(z)=\min\{\ell(z),k\}$ for $z\notin\Delta$.

Step 4 (Clause 1). For $k\in\mathbb{N}$ let $u_{k}=\ell_{k}\,\mathbf{1}_{\mathbb{R}^{2}\setminus\Delta}$, which is Borel by Steps 2 and 3(i) and claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions. If $z\in\Delta$, then $u_{k}(z)=0=\ell(z)$ for every $k$. If $z\notin\Delta$, claim 1 of The Archimedean Property of the Real Numbers gives $k_{0}\in\mathbb{N}$ with $\ell(z)<k_{0}$, and for every $k\ge k_{0}$ Step 3(ii) gives $u_{k}(z)=\min\{\ell(z),k\}=\ell(z)$. In either case the sequence $(u_{k}(z))_{k\in\mathbb{N}}$ is eventually equal to $\ell(z)$ and therefore converges to $\ell(z)$. By claim 5 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, $\ell$ is Borel. Together with Step 2 this proves clause 1.

Step 5 (Clause 2). Let $x,y\in\mathbb{R}$. If $x=y$, then $\ell(x,y)=0$, while $-|x|-|y|\le0$ by claim 1 of Properties of the Absolute Value in an Ordered Field. If $x\ne y$, put $t=|x-y|$, which is positive by claim 1 of Properties of the Absolute Value in an Ordered Field. By Step 1(c), $\ell(x,y)=-\log t\ge-t$, and by claims 5 and 2 of Properties of the Absolute Value in an Ordered Field, $t=|x+(-y)|\le|x|+|-y|=|x|+|y|$. Hence $\ell(x,y)\ge-|x|-|y|$.

Step 6 (Clause 3). Let $\mu\in\mathcal{P}(\mathbb{R})$ satisfy $\mu(\{x\})=0$ for every $x\in\mathbb{R}$. The indicator $\mathbf{1}_{\Delta}$ is a nonnegative Borel function (Step 2), and its integral against $\mu\boxtimes\mu$ is $(\mu\boxtimes\mu)(\Delta)$ by Simple Function and Its Integral and Lebesgue Integral of a Nonnegative Measurable Function. By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product, $\mathbf{1}_{\Delta}\circ\iota$ is measurable for $\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R})$ and

the second equality by the Tonelli part of Tonelli and Fubini Theorems, probability measures being $\sigma$-finite. For fixed $x$, $\mathbf{1}_{\Delta}(x,y)=1$ exactly when $y=x$, so the inner integrand is the indicator of $\{x\}$, which belongs to $\mathcal{B}(\mathbb{R})$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §finite-sets; the inner integral is therefore $\mu(\{x\})=0$. The outer integral is the integral of the zero function, which is $0$. Hence $(\mu\boxtimes\mu)(\Delta)=0$.