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, and we show the condition is also necessary on bounded domains.

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。我们给出了一个熵加权的Orlicz条件：对于单个超线性函数$Ψ$，有$\sup_n \int f_n\, Ψ(|\log f_n|) < \infty$，这严格弱于Godavarti和Hero (2004) 的固定$α$条件。我们还否定了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，并证明了该条件在有界域上也是必要的。