A Probability Measure on the Real Line with a Continuously Differentiable Density of Bounded Derivative Has Finite Free Fisher Information and Finite Logarithmic Energy

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. Lebesgue measure $\lambda$ on $\mathbb{R}=\mathbb{R}^{1}$ is that of Euclidean Space and Lebesgue Measure: Standing Notation §measure, differentiability of a function $\mathbb{R}\to\mathbb{R}$ is that of Derivative at an Interior Point, and Borel is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps. The set $\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$ of measures of finite free Fisher information and the set $\mathcal{D}_{\log}$ of The Logarithmic Energy of a Probability Measure on the Real Line §energy are those of those definitions.

(Smooth densities)¶ Let $\mu\in\mathcal{P}_{2}(\mathbb{R})$ and let $\rho:\mathbb{R}\to\mathbb{R}$ be nonnegative and Borel with $\mu(B)=\int_{B}\rho\,d\lambda$ for every Borel set $B\subseteq\mathbb{R}$. Suppose that $\rho$ is differentiable at every point of $\mathbb{R}$ with a continuous derivative $\rho'$, and that there is $L\in\mathbb{R}$ with $|\rho'(x)|\le L$ for every $x\in\mathbb{R}$. Then $\mu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$ and $\mu\in\mathcal{D}_{\log}$.