We show that under convergence in measure of probability density functions, differential entropy converges whenever the entropy integrands $f_n |\log f_n|$ are uniformly integrable and tight -- a direct consequence of Vitali's convergence theorem. We give an entropy-weighted Orlicz condition: $\sup_n \int f_n\, Ψ(|\log f_n|) < \infty$ for a single superlinear~$Ψ$, strictly weaker than the fixed-$α$ condition of Godavarti and Hero (2004). We also disprove the Godavarti-Hero conjecture that $α> 1$ could be replaced by $α_n \downarrow 1$. We recover the sufficient conditions of Godavarti--Hero, Piera--Parada, and Ghourchian-Gohari-Amini as corollaries. On bounded domains, we prove that uniform integrability of the entropy integrands is both necessary and sufficient -- a complete characterization of entropy convergence.

翻译：我们证明，在概率密度函数依测度收敛的条件下，当熵被积函数 $f_n |\log f_n|$ 一致可积且紧时，微分熵收敛——这是维塔利收敛定理的直接推论。我们给出一个熵加权的Orlicz条件：对于单个超线性函数~$Ψ$，满足 $\sup_n \int f_n\, Ψ(|\log f_n|) < \infty$，这严格弱于Godavarti和Hero (2004) 的固定 $α$ 条件。我们也否定了Godavarti-Hero关于 $α> 1$ 可被 $α_n \downarrow 1$ 替代的猜想。作为推论，我们恢复了Godavarti–Hero、Piera–Parada以及Ghourchian-Gohari-Amini的充分条件。在有界域上，我们证明了熵被积函数的一致可积性既是必要的也是充分的——这给出了熵收敛的一个完整刻画。