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

Each result cited is universally quantified over the data in its own statement. Throughout, $\nu$ and $\gamma$ are probability measures on the measurable space $(S,\mathcal{B}(S))$, by Borel Measure on a Metric Space, and $d_{\mathbb{R}}$ is the absolute-value metric; for a nonnegative integrable function the integral of Integrable Function and the Lebesgue Integral is its integral as a nonnegative measurable function, its negative part being $0$.

Claim 1. Let $h:S\to\mathbb{R}$ be bounded Lipschitz. By A Lipschitz Map is Uniformly Continuous, $h$ is continuous from $(S,d)$ to $(\mathbb{R},d_{\mathbb{R}})$, so by claim 3 of Borel Measurability and Bounded Integration on a Metric Space it is measurable with respect to $\mathcal{B}(S)$ and the Borel $\sigma$-algebra of $(\mathbb{R},d_{\mathbb{R}})$, which is the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ of the real line by claim 2 of that lemma. Thus $h$ is measurable in the sense of Measurable Function and Real-Valued Measurable Function, and being bounded it is bounded measurable.

Claim 2. By Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §criterion, applied to $(S,\mathcal{B}(S))$, $\nu$, $\gamma$ and $C$, it suffices to show $\Lambda^{\gamma}_{h}(\nu)\le C$ for every bounded measurable $h:S\to\mathbb{R}$. Fix such $h$ and a bound $M$ for it. By Nonnegative Combinations of Two Finite Measures, and the Average of Two Couplings §combination with $\alpha=\nu$, $\beta=\gamma$ and $s=t=1$, the set function $\mu=\nu+\gamma$ is a measure on $(S,\mathcal{B}(S))$, hence a Borel measure on $(S,d)$, with $\mu(S)=2$, and

$$(\ddagger)\qquad\int_{S}u\,d\mu=\int_{S}u\,d\nu+\int_{S}u\,d\gamma\quad\text{for every measurable }u:S\to[0,\infty].$$

Let $\varepsilon$ be a positive real; the parameters are chosen in the order $\varepsilon$, then $\delta$ and $\sigma$, then $\eta$, then the functions $\ell_{i}$ and sets $N_{i}$. Let $\delta=\varepsilon/4$. The function $g=h+M$ is measurable by claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and satisfies $0\le g\le2M$, so Approximation of Measurable Functions by Simple Functions §bounded, for the measure space $(S,\mathcal{B}(S),\mu)$ and $K=2M$, gives (choosing an index $m$ with $2^{-m}\le\delta$) a nonnegative simple function $\sigma$ with $0\le g-\sigma\le\delta$ on $S$. Let $\sigma=\sum_{i=1}^{r}c_{i}\mathbf{1}_{B_{i}}$ be its standard representation of Simple Function and Its Integral, with $r\ge1$ (as $S$ is nonempty), $c_{i}\ge0$ and $B_{i}\in\mathcal{B}(S)$. Let $\eta=\varepsilon/(4(M+1)r)$. For each $i$, Inner and Outer Regularity of a Finite Borel Measure on a Metric Space, and Lipschitz Approximation of Indicators §lipschitz, applied to $\mu$, $B_{i}$ and $\eta$, gives a map $\ell_{i}:S\to\mathbb{R}$, Lipschitz with some constant $L_{i}$, with $0\le\ell_{i}\le1$, and a set $N_{i}\in\mathcal{B}(S)$ with $\mu(N_{i})\le\eta$ and $\ell_{i}=\mathbf{1}_{B_{i}}$ on $S\setminus N_{i}$. Let $p=-M+\sum_{i=1}^{r}c_{i}\ell_{i}$ and $k=\max(-M,\min(M,p))$. By the triangle inequality $|p(x)-p(x')|\le(\sum_{i}c_{i}L_{i})\,d(x,x')$ for $x,x'\in S$, and the map $v\mapsto\max(-M,\min(M,v))$ satisfies $|\max(-M,\min(M,v))-\max(-M,\min(M,w))|\le|v-w|$ for real $v,w$; hence $k$ is Lipschitz with constant $\sum_{i}c_{i}L_{i}$ for $d$ and $d_{\mathbb{R}}$ by Lipschitz Map Between Metric Spaces, and $|k|\le M$, so $k$ is bounded Lipschitz and $\Lambda^{\gamma}_{k}(\nu)\le C$ by hypothesis. Let $N=\bigcup_{i=1}^{r}N_{i}\in\mathcal{B}(S)$; by Basic Properties of a Measure §subadditivity (extended by empty sets) $\mu(N)\le r\eta\le\varepsilon/(4(M+1))$. For $x\in S\setminus N$, $p(x)=-M+\sigma(x)$, so $|p(x)-h(x)|=|\sigma(x)-g(x)|\le\delta$, and since $-M\le h(x)\le M$ gives $h(x)=\max(-M,\min(M,h(x)))$, also $|k(x)-h(x)|\le|p(x)-h(x)|\le\delta$; for $x\in N$, $|k(x)-h(x)|\le2M$. Thus $|k-h|\le\delta\mathbf{1}_{S}+2M\mathbf{1}_{N}$ pointwise, where $|k-h|$ is measurable by Claim 1 and claims 2 and 4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, and Linearity and Monotonicity of the Lebesgue Integral §nonnegative with The Integral of an Indicator Function is the Measure of the Set gives

$$\int_{S}|k-h|\,d\mu\le2\delta+2M\mu(N)\le\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon ,$$

so by $(\ddagger)$ both $\int_{S}|k-h|\,d\nu\le\varepsilon$ and $\int_{S}|k-h|\,d\gamma\le\varepsilon$. By Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §functional, $h$ and $k$ are $\nu$-integrable and $\exp\circ h$, $\exp\circ k$ are $\gamma$-integrable, and the nonnegative measurable function $|k-h|$, having finite integral with respect to $\nu$ and to $\gamma$, is $\nu$-integrable and $\gamma$-integrable by Integrable Function and the Lebesgue Integral. Since $h-k\le|k-h|$ pointwise, the linearity and monotonicity of Linearity and Monotonicity of the Lebesgue Integral §integrable give $\int_{S}h\,d\nu-\int_{S}k\,d\nu=\int_{S}(h-k)\,d\nu\le\int_{S}|k-h|\,d\nu\le\varepsilon$, that is $\int_{S}h\,d\nu\le\int_{S}k\,d\nu+\varepsilon$. For reals $a\ge b$ in $[-M,M]$, The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §exp gives $\exp(b-a)\ge1+b-a$, so by claim 1 of Basic Properties of the Exponential Function $\exp(a)-\exp(b)=\exp(a)(1-\exp(b-a))\le\exp(M)(a-b)$; hence $|\exp\circ k-\exp\circ h|\le\exp(M)|k-h|$ pointwise. Writing $Z=\int_{S}\exp\circ h\,d\gamma$ and $Z_{k}=\int_{S}\exp\circ k\,d\gamma$, the linearity and monotonicity of Linearity and Monotonicity of the Lebesgue Integral §integrable, applied to the $\gamma$-integrable functions $\exp\circ k$, $\exp\circ h$ and $|k-h|$ and to the pointwise inequality $\exp\circ k-\exp\circ h\le\exp(M)|k-h|$, give

$$Z_{k}-Z=\int_{S}(\exp\circ k-\exp\circ h)\,d\gamma\le\exp(M)\int_{S}|k-h|\,d\gamma\le\exp(M)\varepsilon ,$$

and $Z\ge\exp(-M)\gamma(S)=\exp(-M)$ because $\exp\circ h\ge\exp(-M)$ ($\exp$ being increasing by claim 4 of Basic Properties of the Exponential Function). By The Natural Logarithm and the inequality $\log x\le x-1$ of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log,

$$\log Z_{k}-\log Z=\log\frac{Z_{k}}{Z}\le\frac{Z_{k}-Z}{Z}\le\exp(2M)\varepsilon .$$

Therefore $\Lambda^{\gamma}_{h}(\nu)=\int_{S}h\,d\nu-\log Z\le\int_{S}k\,d\nu+\varepsilon-\log Z_{k}+\exp(2M)\varepsilon=\Lambda^{\gamma}_{k}(\nu)+(1+\exp(2M))\varepsilon\le C+(1+\exp(2M))\varepsilon$. As $\varepsilon$ was an arbitrary positive real, $\Lambda^{\gamma}_{h}(\nu)\le C$.

Claim 3. Let $h:S\to\mathbb{R}$ be bounded Lipschitz; it is bounded measurable by Claim 1 and continuous by A Lipschitz Map is Uniformly Continuous. Let $Z=\int_{S}\exp\circ h\,d\gamma$, a positive real by Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §functional. For each $j$, Relative Entropy on a Measurable Space: the Gibbs Inequality, the Variational Criterion and Formula, the Entropy Inequality, Small Sets and Data Processing §gibbs, applied to $(S,\mathcal{B}(S))$, $\nu_{j}$ and $\gamma$, gives

$$\int_{S}h\,d\nu_{j}-\log Z=\Lambda^{\gamma}_{h}(\nu_{j})\le H(\nu_{j}\,|\,\gamma)\le C .$$

Since $\nu_{j}\Rightarrow\nu$ and $h$ is bounded and continuous, $\int_{S}h\,d\nu_{j}\to\int_{S}h\,d\nu$ by Weak Convergence of Finite Borel Measures on a Metric Space, so passing to the limit gives $\Lambda^{\gamma}_{h}(\nu)=\int_{S}h\,d\nu-\log Z\le C$. This holds for every bounded Lipschitz $h$, so Claim 2 shows that $\nu$ has finite relative entropy with respect to $\gamma$ and $H(\nu\,|\,\gamma)\le C$.