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 ⊂ 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 pintegrable upper gradients, about p-quasiopen, p-path and p-finely open sets, and about Lebesgue points for N 1,1 -functions, to spaces that only satisfy local assumptions.