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

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, in dimension $d=1$, with the real line identified with $\mathbb{R}^{1}$ and $\mathbb{R}^{2}=\mathbb{R}^{1+1}$ as in One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative, so that $\mu\boxtimes\mu\in\mathcal{P}(\mathbb{R}^{2})$ is the product measure of $\mu\in\mathcal{P}(\mathbb{R})$ with itself. $\log$ is the natural logarithm on the positive reals, $|s|$ the absolute value of $s\in\mathbb{R}$, Borel for maps is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, and a subset of $\mathbb{R}^{2}$ is Borel when it belongs to $\mathcal{B}(\mathbb{R}^{2})$, as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces. Let $\ell:\mathbb{R}^{2}\to\mathbb{R}$ be given by

and let $\Delta=\{(x,x):x\in\mathbb{R}\}$ be the diagonal of $\mathbb{R}^{2}$.

1. (Borel measurability)¶ $\ell$ is Borel, and $\Delta$ is a Borel subset of $\mathbb{R}^{2}$.

2. (Lower bound)¶ $-|x|-|y|\le\ell(x,y)$ for all $x,y\in\mathbb{R}$.

3. (Null diagonal)¶ If $\mu\in\mathcal{P}(\mathbb{R})$ satisfies $\mu(\{x\})=0$ for every $x\in\mathbb{R}$, then $(\mu\boxtimes\mu)(\Delta)=0$.