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

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. The set $\mathcal{D}_{\log}$ and the logarithmic energy $\mathcal{E}_{\log}$ are those of that definition; $M_{2}$ is the second moment; the translations $\tau_{a}$ and push-forwards are those of that setting and of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward; lower semicontinuity on $\mathcal{D}_{\log}$ is taken relative to $\mathcal{D}_{\log}$ in the metric space $(\mathcal{P}_{2}(\mathbb{R}),W_{2})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §measures; and limits of real sequences are those of that definition.

1. (Lower bound)¶ $-\bigl(1+M_{2}(\mu)\bigr)\le\mathcal{E}_{\log}(\mu)$ for every $\mu\in\mathcal{D}_{\log}$.

2. (Translation invariance)¶ For $\mu\in\mathcal{D}_{\log}$ and $a\in\mathbb{R}$, $(\tau_{a})_{\#}\mu\in\mathcal{D}_{\log}$ and $\mathcal{E}_{\log}((\tau_{a})_{\#}\mu)=\mathcal{E}_{\log}(\mu)$.

3. (Closed sublevel sets)¶ Let $\mu_{n}\in\mathcal{D}_{\log}$ for $n\in\mathbb{N}$, $\mu\in\mathcal{P}_{2}(\mathbb{R})$ and $c\in\mathbb{R}$ satisfy $\mathcal{E}_{\log}(\mu_{n})\le c$ for every $n$, with $(W_{2}(\mu_{n},\mu))_{n\in\mathbb{N}}$ of limit $0$. Then $\mu\in\mathcal{D}_{\log}$ and $\mathcal{E}_{\log}(\mu)\le c$.

4. (Lower semicontinuity)¶ $\mathcal{E}_{\log}$ is lower semicontinuous on $\mathcal{D}_{\log}$.