A uniqueness result for functions with zero fine gradient on quasiconnected and finely connected sets

We show that every Sobolev function in $W_{\text{loc}}^{1,p}(U)$ on a $p$-quasiopen set $U\subset {\mathbf{R}}^n$ with a.e.-vanishing $p$-fine gradient is a.e.-constant if and only if $U$ is $p$-quasiconnected. To prove this we use the theory of Newtonian Sobolev spaces on metric measure spaces, and obtain the corresponding equivalence also for complete metric spaces equipped with a doubling measure supporting a $p$-Poincar'e inequality. On unweighted ${\bf R}^n$,wealsoobtainthecorrespondingresultfor$p$-finely open sets in terms of $p$-fine connectedness, using a deep result by Latvala.