Density-degree function for subsets of Rn

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.