A SHORT NOTE ON ENHANCED DENSITY SETS

We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113(3) (2004), 397–401) result: ‘If Ω is a set of locally finite perimeter in ℝ2, then there is no function f ∈ C1(ℝ2) such that ∇f(x1, x2) = (x2, 0) at...

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Published inGlasgow mathematical journal Vol. 53; no. 3; pp. 631 - 635
Main Author DELLADIO, SILVANO
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.09.2011
Subjects
Approximation
Density
Mathematical analysis
Mathematics
Proving
Primary 28Axx
28A75
Secondary 26A45
49Q15
Online AccessGet full text
ISSN0017-0895
1469-509X
DOI10.1017/S001708951100022X

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Summary:We give a simple proof of a statement extending Fu's (J.H.G. Fu, Erratum to ‘some remarks on legendrian rectiable currents’, Manuscripta Math. 113(3) (2004), 397–401) result: ‘If Ω is a set of locally finite perimeter in ℝ2, then there is no function f ∈ C1(ℝ2) such that ∇f(x1, x2) = (x2, 0) at a.e. (x1, x2) ∈ Ω’. We also prove that every measurable set can be approximated arbitrarily closely in L1 by subsets that do not contain enhanced density points. Finally, we provide a new proof of a Poincaré-type lemma for locally finite perimeter sets, which was first stated by Delladio (S. Delladio, Functions of class C1 subject to a Legendre condition in an enhanced density set, to appear in Rev. Mat. Iberoamericana).
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ISSN:0017-0895
1469-509X
DOI:10.1017/S001708951100022X