The Logarithmic Energy of a Probability Measure on the Real Line

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 identifications of 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, and integrable is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures. Let $\ell:\mathbb{R}^{2}\to\mathbb{R}$ be the logarithmic kernel of that lemma, which 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.

(Logarithmic energy)¶ The set $\mathcal{D}_{\log}$ consists of those $\mu\in\mathcal{P}_{2}(\mathbb{R})$ for which $\mu(\{x\})=0$ for every $x\in\mathbb{R}$ and $\ell$ is $\mu\boxtimes\mu$-integrable. For $\mu\in\mathcal{D}_{\log}$ the logarithmic energy of $\mu$ is the real number