Two point sets with additional properties

Bienias, Marek; Głąb, Szymon; Rałowski, Robert; Żeberski, Szymon

doi:10.1007/s10587-013-0069-2

Cited by 3 publications

(2 citation statements)

References 16 publications

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“…It is not known if Ω can satisfy the assumptions in Theorem 1.2 and at the same time be nonmeasurable in X. Nevertheless, in Section 6 we construct a measurable set in R 2 , with full measure and satisfying the conclusions in Theorems 1.1 and 1.2 (except for the last part), but which is not even p-path almost open in R 2 .…”

Section: Introductionmentioning

confidence: 99%

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

Björn¹,

Björn²,

Lahti³

2021

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%

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

Björn¹,

Björn²,

Lahti³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…as small sets in some sense (see [7], [11]) or beacuse they provide existence of other types of sets with certain properties (see [18]). Subsets of the plane in the view of Luzin and Sierpiński sets were also considered recently in [3].…”

Section: Introductionmentioning

confidence: 99%