The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets

In the setting of Optimal Transport on the Flat Torus: Standing Notation, let finite entropy and the entropy $\mathrm{Ent}$ be those of that definition, and ¶write $\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ for the set of members of $\mathcal{P}(\mathbb{T}^{d})$ with finite entropy. Then the following hold.

1. (Nonnegativity and absolute continuity)¶ Every $\mu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ is absolutely continuous and satisfies $\mathrm{Ent}(\mu)\ge0$.

2. (Closed sublevel sets)¶ Let $C$ be a real number, let $\mu\in\mathcal{P}(\mathbb{T}^{d})$, and let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ with $\mathrm{Ent}(\mu_{n})\le C$ for every $n$ that converges to $\mu$ in $(\mathcal{P}(\mathbb{T}^{d}),W_{\mathbb{T}})$. Then $\mu\in\mathcal{P}^{\mathrm{Ent}}(\mathbb{T}^{d})$ and $\mathrm{Ent}(\mu)\le C$.