Addendum to "The Kolmogorov--Riesz compactness theorem" (Expo. Math. 28:385--394 (2010))

We show how to improve on Theorem 10 in [arXiv:0906.4883], describing when subsets in $W^{1,p}(\mathbb{R}^n)$ are totally bounded subsets of $L^q(\mathbb{R}^n)$ for $p<n$ and $p\le q<p^*$. This improvement was first shown by Dosso, Fofana, and Sanogo (2013) in the context of Morrey--Sobolov spaces.