Removable sets for Newtonian Sobolev spaces and a characterization of $p$-path almost open sets

Abstract

We study removable sets for Newtonian Sobolev functions in metric measure spaces satisfying the usual (local) assumptions of a doubling measure and a Poincaré inequality.In particular, when restricted to Euclidean spaces, a closed set $E\subset {\mathbb{R}}^n$ with zero Lebesgue measure is shown to be removable for $W^{1,p}({\mathbb{R}}^n\setminus E)$ if and only if ${\mathbb{R}}^n\setminus E$ supports a $p$-Poincaré inequality as a metric space. When $p>1$, this recovers Koskela’s result (Ark. Mat. $37$ (1999), 291–304), but for $p=1$, as well as for metric spaces, it seems to be new. We also obtain the corresponding characterization for the Dirichlet spaces $L^{1,p}$. To be able to include $p=1$, we first study extensions of Newtonian Sobolev functions in the case $p=1$ from a noncomplete space $X$ to its completion $\hat{X}$. In these results, $p$-path almost open sets play an important role, and we provide a characterization of them by means of $p$-path open, $p$-quasiopen and $p$-finely open sets. We also show that there are nonmeasurable $p$-path almost open subsets of ${\mathbb{R}}^n$, $n\ge 2$, provided that the continuum hypothesis is assumed to be true. Furthermore, we extend earlier results about measurability of functions with $L^p$-integrable upper gradients, about $p$-quasiopen, $p$-path open and $p$-finely open sets, and about Lebesgue points for $N^{1,1}$-functions, to spaces that only satisfy local assumptions.