Some remarks on points of Lebesgue density and density-degree functions

Some properties of m-density points and density-degree functions are studied. Moreover the following main results are provided: 2mm {itemize} { Let λ be a continuous differential form of degree h in Rⁿ (with h≥ 0) having the following property: There exists a continuous differential form Δ of degree $h+1$ in ⁿ such that {equation*} ∫_{{ R}^n}Δω =∫_{{ R}^n}λ dω, {equation*} for every C^∞c differential form ω of degree $n-h-1$ in Rⁿ. Moreover let μ be a C¹ differential form of degree $h+1$ in Rⁿ and set E:=∈ Rⁿ\,\, Δ (y)=μ(y)\. Then dμ (x) = 0 whenever x is a $(n+1)$-density point of E.} 2mm { Let f: Rⁿ→ R̄ be a measurable function such that f(x)∈ \0\∪ [n,+∞] for a.e. x∈ Rⁿ. Then there exists a countable family ₖ\ₖ₌₁^∞ of closed subsets of Rⁿ such that the corresponding sequence of density-degree functions ₖ\ₖ₌₁^∞ converges almost everywhere to f. }