We define $U : [0, +\infty) \to [0, +\infty)$ by $U(0) := 0$ and $U (s) := s \log s$ for $s >0$. Then $U$ is strictly convex. Let $D$ be the set of all bounded non-negative continuous functions $\rho : \mathbb R^d \to \mathbb R$ such that

- $\int_{\mathbb R^d} \rho=1$.
- $M (\rho) :=\int_{\mathbb R^d} |x|^2 \rho (x) \, \mathrm d x < +\infty$
- $H(\rho) := \int_{\mathbb R^d} U ( \rho (x)) \, \mathrm d x< +\infty$.

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

Assume that $\rho, \rho_n \in D$ such that $\|\rho_n - \rho\|_{\infty} \to 0$ as $n \to \infty$. Is it true that $|H(\rho_n) - H(\rho)| \to 0$ as $n \to \infty$?

Thank you so much for your elaboration!