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Let $\mathcal H (\mu)$ be the relative entropy of $\mu$ w.r.t. Lebesgue measure. It is well-known that $\mathcal H$ is not continuous but only lower semi-continuous w.r.t. $W_2$. …

We define the entropy $\mathcal H : \mathcal P_2^a \to [0, \infty]$ by $$ \mathcal H (\rho) := \int_{\mathbb R^d} U \circ \rho. $$ Then $\mathcal H (\rho) > 0$ for all $\rho \in \mathcal P_2^a$. …

If $\rho \in D$ then $\rho$ is a probability density function whose induced measure has finite second moment and finite Boltzmann entropy. …

As such, Boltzmann entropy is well-defined (in extended real number line) for every $f \in D_2$. …

L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let $$ \mathcal{H}(\rho)=\int_{\mathbb{R}^d}\rho\log \rho \,dx $$ be the Boltzmann entropy …