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

### Anders Björn

Linköping University, Sweden### Jana Björn

Linköping University, Sweden### Panu Lahti

Chinese Academy of Sciences, Beijing, China

## 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 $E⊂R_{n}$ with zero Lebesgue measure is shown to be removable for $W_{1,p}(R_{n}∖E)$ if and only if $R_{n}∖E$ 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 $X^$. 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 $R_{n}$, $n≥2$, 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.

## Cite this article

Anders Björn, Jana Björn, Panu Lahti, Removable sets for Newtonian Sobolev spaces and a characterization of $p$-path almost open sets. Rev. Mat. Iberoam. 39 (2023), no. 3, pp. 1143–1180

DOI 10.4171/RMI/1419