Relative Entropy on a Metric Space: the Variational Criterion over Bounded Lipschitz Functions and Sequentially Closed Sublevel Sets under Weak Convergence

In the settings of The Real Numbers: Standing Notation and Background and Measure Spaces and the Lebesgue Integral: Standing Notation, the measure space of the latter being instantiated at each use by the measure space named, let $(S,d)$ be a metric space with $S$ nonempty and Borel $\sigma$-algebra $\mathcal{B}(S)$, and let $\nu$ and $\gamma$ be Borel measures on $(S,d)$ with $\nu(S)=\gamma(S)=1$. Finite relative entropy with respect to $\gamma$ and $H(\nu\,|\,\gamma)$ are those of that definition on $(S,\mathcal{B}(S))$; bounded measurable functions and the numbers $\Lambda^{\gamma}_{h}(\nu)$ are those of Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing for the measurable space $(S,\mathcal{B}(S))$. A function $h:S\to\mathbb{R}$ is called bounded Lipschitz if it is bounded and Lipschitz with a constant for $d$ and the absolute-value metric; by claim 1 below and Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §functional, $\Lambda^{\gamma}_{h}(\nu)$ is a real number for every bounded Lipschitz $h$. Weak convergence of Borel measures on $(S,d)$ is written $\nu_{j}\Rightarrow\nu$.

1. (Lipschitz functions)¶ Every bounded Lipschitz function $h:S\to\mathbb{R}$ is bounded measurable.

2. (Lipschitz criterion)¶ Let $C\in\mathbb{R}$ be such that $\Lambda^{\gamma}_{h}(\nu)\le C$ for every bounded Lipschitz $h:S\to\mathbb{R}$. Then $\nu$ has finite relative entropy with respect to $\gamma$, and $H(\nu\,|\,\gamma)\le C$.

3. (Sequentially closed sublevel sets)¶ Let $C\in\mathbb{R}$ and let $(\nu_{j})_{j\in\mathbb{N}}$ be a sequence of Borel measures on $(S,d)$ with $\nu_{j}(S)=1$, each of finite relative entropy with respect to $\gamma$ with $H(\nu_{j}\,|\,\gamma)\le C$, such that $\nu_{j}\Rightarrow\nu$. Then $\nu$ has finite relative entropy with respect to $\gamma$, and $H(\nu\,|\,\gamma)\le C$.