For all subsets E of Rn, we define a function dE measuring the density-degree of E at the points of Rn. We provide some results which involve dE. In particular we prove an approximation property stating that, given a bounded open set Ω, the following facts hold: • For all C &lt; Ln(Ω) there exists a closed subset F of Ω such that Ln(F) &gt; C and dF = n almost everywhere in F; • For all C &lt; Ln(Ω) and for every proper subinterval I of (n, +∞), there exists a closed subset F of Ω and an open subset U of Ω such that F ⊃ Ω\U, Ln(U) &lt; Ln(Ω)−C (hence Ln(F) &gt; C) and dF (x) ∈ I for all x ∈ Ω U.

Density-degree function for subsets of Rn / Delladio, S.. - In: HOUSTON JOURNAL OF MATHEMATICS. - ISSN 0362-1588. - STAMPA. - 45:3(2019), pp. 743-762.

### Density-degree function for subsets of Rn

#### Abstract

For all subsets E of Rn, we define a function dE measuring the density-degree of E at the points of Rn. We provide some results which involve dE. In particular we prove an approximation property stating that, given a bounded open set Ω, the following facts hold: • For all C < Ln(Ω) there exists a closed subset F of Ω such that Ln(F) > C and dF = n almost everywhere in F; • For all C < Ln(Ω) and for every proper subinterval I of (n, +∞), there exists a closed subset F of Ω and an open subset U of Ω such that F ⊃ Ω\U, Ln(U) < Ln(Ω)−C (hence Ln(F) > C) and dF (x) ∈ I for all x ∈ Ω U.
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2019
3
Density-degree function for subsets of Rn / Delladio, S.. - In: HOUSTON JOURNAL OF MATHEMATICS. - ISSN 0362-1588. - STAMPA. - 45:3(2019), pp. 743-762.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11572/294811`