A closure result for differential forms in the context of superdensity, approximation of measurable functions by density-degree functions

Some properties of m-density points and density-degree functions are studied. Moreover, the following main results are provided: Let λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document} be a continuous differential form of degree h in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} (with h≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\ge 0$$\end{document}) having the following property: there exists a continuous differential form Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta$$\end{document} of degree h+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h+1$$\end{document} in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} such that ∫RnΔ∧ω=∫Rnλ∧dω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{{\mathbb {R}}^n}\Delta \wedge \omega =\int _{{\mathbb {R}}^n}\lambda \wedge \mathrm{{d}}\omega , \end{aligned}$$\end{document} for every Cc∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty _c$$\end{document} differential form ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega$$\end{document} of degree n-h-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-h-1$$\end{document} in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document}. Moreover let μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} be a C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} differential form of degree h+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h+1$$\end{document} in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} and set E:={y∈Rn|Δ(y)=μ(y)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E:=\{y\in {\mathbb {R}}^n\,\vert \, \Delta (y)=\mu (y)\}$$\end{document}. Then dμ(x)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{{d}}\mu (x) = 0$$\end{document} whenever x is a (n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+1)$$\end{document}-density point of E. Let f:Rn→R¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:{\mathbb {R}}^n\rightarrow {\overline{{\mathbb {R}}}}$$\end{document} be a measurable function such that f(x)∈{0}∪[n,+∞]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)\in \{0\}\cup [n,+\infty ]$$\end{document} for a.e. x∈Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in {\mathbb {R}}^n$$\end{document}. Then there exists a countable family {Fk}k=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{F_k\}_{k=1}^\infty$$\end{document} of closed subsets of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} such that the corresponding sequence of density-degree functions {dFk}k=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{d_{F_k}\}_{k=1}^\infty$$\end{document} converges almost everywhere to f. Let λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda$$\end{document} be a continuous differential form of degree h in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} (with h≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\ge 0$$\end{document}) having the following property: there exists a continuous differential form Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta$$\end{document} of degree h+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h+1$$\end{document} in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} such that ∫RnΔ∧ω=∫Rnλ∧dω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{{\mathbb {R}}^n}\Delta \wedge \omega =\int _{{\mathbb {R}}^n}\lambda \wedge \mathrm{{d}}\omega , \end{aligned}$$\end{document} for every Cc∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty _c$$\end{document} differential form ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega$$\end{document} of degree n-h-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-h-1$$\end{document} in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document}. Moreover let μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} be a C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} differential form of degree h+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h+1$$\end{document} in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} and set E:={y∈Rn|Δ(y)=μ(y)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E:=\{y\in {\mathbb {R}}^n\,\vert \, \Delta (y)=\mu (y)\}$$\end{document}. Then dμ(x)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{{d}}\mu (x) = 0$$\end{document} whenever x is a (n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+1)$$\end{document}-density point of E. Let f:Rn→R¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:{\mathbb {R}}^n\rightarrow {\overline{{\mathbb {R}}}}$$\end{document} be a measurable function such that f(x)∈{0}∪[n,+∞]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)\in \{0\}\cup [n,+\infty ]$$\end{document} for a.e. x∈Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in {\mathbb {R}}^n$$\end{document}. Then there exists a countable family {Fk}k=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{F_k\}_{k=1}^\infty$$\end{document} of closed subsets of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document} such that the corresponding sequence of density-degree functions {dFk}k=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{d_{F_k}\}_{k=1}^\infty$$\end{document} converges almost everywhere to f.


Introduction
Let m ∈ [n, +∞) and E ⊂ ℝ n . Then x ∈ ℝ n is said to be an m-density point of E if L n (B(x, r)⧵E) = o(r m ) as r → 0 + , where L n denotes the Lebesgue outer measure in ℝ n and B(x, r) is the open ball of radius r centered at x. The set of m-density points of E is denoted by E (m) (cf. [2,3]). For x ∈ E (n) , the supremum of the numbers m such that x ∈ E (m) is called the density-degree of E at x and is denoted by d E (x) , while d E (x) is assumed to be zero when x ∉ E (n) (cf. [7]). Thanks to these definitions, points of Lebesgue density cease to be indistinguishable from each other and are instead characterised by their own density-degree.
There are reasons for considering points with a high density-degree, albeit finite, as interior points. For example, the following fact holds: If x ∈ {gradf = F} (n+1) , where f ∈ C 1 (ℝ n ) and F ∈ C 1 (ℝ n , ℝ n ) , then the Jacobian matrix DF(x) is symmetric. We thus discover that the geometry of {gradf = F} is characterised by a very low density at points where DF is non-symmetric (cf. [3,Theorem 2.1]; subsequent extensions to the context of PDE can be found e.g. in [9,Theorem 3.6] and [11,Theorem 3.2

]).
A similar application can be given in the context of the Frobenius theorem on distributions. More precisely let M and D be, respectively, a n-dimensional C 1 submanifold of ℝ n+k and a C 1 distribution of rank n on ℝ n+k . Then D is involutive at each (n + 1)-density point of the tangency set of M with respect to D. Hence the tangency must be low in density at all points where the distribution D is noninvolutive (cf. [8,Theorem 1.1]).
Another fact worth mentioning in this introduction is the following: Except for a subset of null measure, the points of a set of locally finite perimeter are m-density points, with m ∶= n + 1 + 1 n−1 (cf. [3,Lemma 4.1]; for further applications in this context, see [5,6]).
Using m-density one can also define the notion of m-approximate continuity, which for m = n reduces to the well known approximate continuity (cf. [13,Section 2.9.12] and [12,Section 1.7.2]). Properties of m-approximate continuity holds for Sobolev functions and for functions of bounded variation (cf. [10]).
Finally, let us briefly describe the two main results of this note. The first one, proved in Sect. 3 below, is the following generalization of [3, Theorem 2.1].
Theorem Let be a continuous differential form of degree h in ℝ n (with h ≥ 0 ) having the following property: there exists a continuous differential form Δ of degree h + 1 in ℝ n such that for every C ∞ c differential form of degree n − h − 1 in ℝ n . Moreover let be a C 1 differential form of degree h + 1 in ℝ n and set E ∶= {y ∈ ℝ n | Δ(y) = (y)} . Then The second result is provided in Sect. 5. It originates from the following question: Is it true that for every (measurable) function f ∶ ℝ n → {0} ∪ [n, +∞] there exists E ⊂ ℝ n such that d E = f almost everywhere? Some hasty considerations may mislead us into thinking that the answer is yes, but on deeper reflection, it is not difficult to conclude that the correct answer is no (cf. Example 5.1 below). However, somewhat surprisingly, the following approximation property holds true.

General notation
The coordinates of ℝ n are denoted by (x 1 , … , x n ) and we set D i ∶= ∕ x i . If k is any positive integer not exceeding n, then I(n, k) is the family of integer multi-indices If ∈ I(n, k) , then we denote by ̄ the member of I(n, n − k) which complements in {1, 2, … , n} in the natural increasing order (e.g., if = (2, 3, 5) ∈ I(7, 3) , then ̄= (1, 4, 6, 7) ∈ I (7,4) ). The open ball of radius r centered at x ∈ ℝ n is denoted by B(x, r). Sometimes, for simplicity, the sphere B(0, r) will be denoted by B r . Moreover E is the characteristic function of E. The equivalence relation of functions and the equivalence relation of subsets of ℝ n , with respect to the Lebesgue outer measure L n , are both denoted by ∼ . If E, F ⊂ ℝ n and L n (E⧵F) = 0 , then we write

Covectors and differential forms in ℝ n
If k is a positive integer not exceeding n, then a k-covector (of ℝ n ) is a k-linear alternating map from (ℝ n ) k to ℝ . Let ⋀ k (ℝ n ) denote the set of all k-covectors. In particular ⋀ 1 (ℝ n ) is the dual space of ℝ n and we will denote the standard dual basis by where ∧ denotes the wedge product. Recall that ⋀ k (ℝ n ) is equipped with the following inner product naturally induced from ℝ n (making (1) an orthonormal basis): We observe that the following property holds (we set for simplicity dx ∶= dx 1 ∧ ⋯ ∧ dx n ): dx , then for all ∈ I(n, k) we have 0 = ⟨ ∧ dx̄, dx⟩ = ⟨dx ∧ dx̄, dx⟩, hence = 0.
A C H differential form of degree k (on ℝ n ) is a k-covector field with {f | ∈ I(n, k)} ⊂ C H (ℝ n ) . A C H differential form of degree 0 is simply a function in C H (ℝ n ) . We recall that the addition and the exterior product of covectors naturally induce the addition and the exterior product of differential forms. Moreover an exterior derivative operator d is uniquely defined on C 1 differential forms and it holds that • On C 1 differential forms of degree 0, the operator d agrees with the ordinary differential on C 1 (ℝ); • d( + ) = d + d , for all C 1 differential forms and of degree k; • d( ∧ ) = d ∧ + (−1) k ∧ d , for all C 1 differential forms and , if has degree k; • d(d ) = 0 , for all C 2 differential forms .
We also recall that, given a continuous differential form of degree n with compact support and a measurable set E ⊂ ℝ n , the integral of on E is defined as follows: For a comprehensive treatment of k-covectors and differential forms, we refer the reader to the numerous books on differential geometry and geometric measure theory that deal with this subject, e.g., [15,16].

Definition 2.1
Let m ∈ [n, +∞) and E ⊂ ℝ n . Then x ∈ ℝ n is said to be a "m-density The set of m-density points of E is denoted by E (m) .

Remark 2.2
The following simple facts occur: be any family of subsets of ℝ n and m ∈ [n, +∞).

Theorem 2.4 ([7], Corollary 4.1) If E is a measurable subset of ℝ n and m ∈ (n, +∞) , then
The Lebesgue density theorem states that if E is a measurable subset of ℝ n , then almost every x ∈ E is a n-density point of E. A remarkable family of sets that turn out to be strictly more dense than generic measurable sets is that of finite perimeter sets. We recall that the perimeter of a measurable set E ⊂ ℝ n , denoted by P(E), is the variation of E , that is In the special case when E is of class C 1 , the perimeter P(E) agrees with the natural (n − 1)-dimensional hypersurface measure of E . An excellent account of finite perimeter sets can be found, e.g., in [1,14]. The following results show that the order of density of every finite perimeter set is not less than the number and more precisely that m 0 is the maximum order of density common to all sets of finite perimeter.

The density-degree function
Prior to providing the definition of the density-degree function, observe that if E is a subset of ℝ n and x ∈ ℝ n , then the set {k ∈ [n, +∞) | x ∈ E (k) } is a (possibly empty) interval.

Definition 2.7
Let E be a subset of ℝ n . Then define the "density-degree function of When there exists k ∈ [n, +∞] such that we say that E is a "uniformly k-dense set".
Observe that strict inclusion can occur, e.g. E ∶= B r ⧵{0} (for which one has d −1

Example 2.9
Let m ∈ (2, +∞) and E be the set of points (x 1 , x 2 ) ∈ ℝ 2 satisfying |x 2 | > |x 1 | m−1 . Since (as an elementary computation shows) This proposition collects a number of properties which have been proved in Proposition 5.1 and Proposition 5.2 of [7] (except (7) which is very easy to verify).

Proposition 2.10
Let E be a subset of ℝ n and m ∈ [n, +∞) . The following properties hold:

Remark 2.11
Both the inclusions in statement (4) of Proposition 2.10 may be strict (cf. Proposition 2.13 below and Example 2.9, respectively).

Remark 2.12
Let Ω and E be, respectively, an open subset of ℝ n and a measurable subset of Ω . We observe that the existence of even a single point x ∈ Ω such that d E (x) < +∞ yields L n (Ω⧵E) > 0 . This simple observation might lead us to believe that L n (E) must be small if the set of such points x has a large measure. Proposition 2.13 and Theorem 2.14 below show, in particular, that this is not true.
The following result establishes that a bounded open set in ℝ n can be arbitrarily approximated from inside by closed uniformly n-dense sets.
We expect that Proposition 2.13 can be extended to a result of approximation from inside by closed uniformly k-dense sets, for all k ≥ n . We are not yet able to resolve this conjecture, but we have the following result.
In particular, one has t ≤ d F (x) ≤ m for all x ∈ Ω⧵U.

Remark 2.15
When Ω is Lipschitz, it is obvious that the closed set F in Proposition 2.13 and in Theorem 2.14 can be chosen so that F ⊂ Ω.
Finally, observe that Theorem 2.5 can be restated as follows:

A simple characterization of superdensity points
Proposition 3.1 Let E ⊂ ℝ n be measurable, x ∈ ℝ n and m ∈ [n, +∞) .
(1) If x ∈ E (m) , then for all ∈ C c (ℝ n ) and for every measurable function g ∶ ℝ n → ℝ which is bounded in a neighborhood of x. (2) Let g ∈ C(ℝ n ) be such that g(x) ≠ 0 and (4) holds for all ∈ C c (ℝ n ) . Then x ∈ E (m) .
Proof (1) Let us consider x ∈ E (m) , ∈ C c (ℝ n ) and an arbitrary measurable function g ∶ ℝ n → ℝ which is bounded in a neighborhood of x. If R is a positive number such that supp( ) ⊂ B(0, R) , then Hence (4) follows at once.
(2) Let us consider ∈ C c (ℝ n ) such that Moreover, without loss of generality, we can assume that g is positive in a neighborhood of x. Hence two positive constants p and r 0 have to exist such that inf B(x,2r 0 ) g ≥ p . For all r ∈ (0, r 0 ] we have hence the conclusion follows from (4). ◻ This corollary is a trivial consequence of Proposition 3.1.

Corollary 3.2
Let E ⊂ ℝ n be measurable, x ∈ ℝ n and m ∈ [n, +∞) . Then each of the following properties is equivalent to x ∈ E (m) : (1) The equation holds for all ∈ C c (ℝ n ). (2) There exists g ∈ C(ℝ n ) such that g(x) ≠ 0 and (4) holds for all ∈ C c (ℝ n ).
(3) The identity (4) holds for all ∈ C c (ℝ n ) and for every measurable function g ∶ ℝ n → ℝ which is bounded in a neighborhood of x.
The next result is also a very easy consequence of Proposition 3.1.

Corollary 3.3 Let us consider a measurable set
and a measurable function Γ ∶ ℝ n → ℝ which is continuous at x. Assume that there exists ∈ C c (ℝ n ) such that ∫ ℝ n dL n ≠ 0 and (for r small enough) where g 1 , … , g k ∶ ℝ n → ℝ is a family of measurable functions which are bounded in a neighborhood of x and 1 , … , k ∈ C c (ℝ n ) . Then Γ(x) = 0.

Proof From (4) it follows that
On the other hand, we have also so that Hence (letting r → 0+) and the conclusion follows recalling that ∫ ℝ n dL n ≠ 0 . ◻ We are interested in Corollary 3.3 as it provides a common argument for the proofs of several theorems we have obtained in our work on superdensity, e.g., [3, Theorem 2.1] (the oldest one) and [9, Theorem 3.6] (the most recent one). Having it in mind can be useful for yielding new applications. As an example, let us prove the following result.

Theorem 3.4
Let be a continuous differential form of degree h on ℝ n (with h ≥ 0 ) having the following property: There exists a continuous differential form Δ of degree h + 1 in ℝ n such that for every C ∞ c differential form of degree n − h − 1 in ℝ n . Moreover let be a C 1 differential form of degree h + 1 in ℝ n and set E ∶= {y ∈ ℝ n | Δ(y) = (y)} . Then Proof Let x ∈ E (n+1) . Consider an arbitrary C ∞ differential form of degree n − h − 2 in ℝ n and define Γ ∶ ℝ n → ℝ as Moreover let ∈ C ∞ c (ℝ n ) be such that ∫ ℝ n dL n ≠ 0 . Then, from the Stokes theorem, we obtain (for r small enough) Γ(y) ∶= ⟨d (y) ∧ (y), dx 1 ∧ ⋯ ∧ dx n ⟩ (y ∈ ℝ n ).

Hence
where and Now observe that y − x r dL n (y) = I(r) + J(r),

Some further properties of the density-degree functions
Proposition 4.1 Given two subsets E, F of ℝ n , the following properties hold: Proof (1) Since L n (E ∩ F c ) = L n (F ∩ E c ) = 0 , the sets E ∩ F c and F ∩ E c are measurable. It follows that (for all x ∈ ℝ n and r > 0) where the second term on the right above is 0 by monotonicity. Since the last term is symmetric in E, F, one has hence E (m) = F (m) for all m ≥ n.
(2) First of all, recall that (by a well-known result on the Lebesgue set, e.g., see Corollary 1.5 in [18, Chapter 3]) a measure zero set N ⊂ ℝ n has to exist such that and Moreover, by hypothesis, at least one of the following inequalities must hold: • If the first inequality holds, then, by (9), for all x in the positive measure set ). • If instead the second inequality holds, then, by (10), for all x in the positive ◻ The following property follows immediately from (1) of Proposition 4.1, by also recalling that Lebesgue outer measure is Borel regular. If J is finite, then the first one turns into the equality d ∩ j∈J E j = min j∈J d E j , while the identity d ∪ j∈J E j = max j∈J d E j fails to be true in general.
Proof For all x ∉ (∩ j∈J E j ) (n) we obviously have So let us suppose x ∈ (∩ j∈J E j ) (n) . Then the set {k ≥ n | x ∈ (∩ j∈J E j ) (k) } is non-empty and where equality holds in the second line whenever J is finite (cf. Remark 2.2). This concludes the proof of the part concerning d ∩ j∈J E j .
For all x ∉ (∪ j∈J E j ) (n) , one also has x ∉ E (n) j for all j ∈ J (cf. Remark 2.2) and thus On the other hand, if x ∈ (∪ j∈J E j ) (n) , then the set {k ≥ n | x ∈ (∪ j∈J E j ) (k) } is nonempty and (cf. Remark 2.2) To verify the last assertion, we can consider the example provided in Remark 2.2 ( n = 1 , J = {1, 2} , E 1 = (−1, 0) , E 2 = (0, 1) ) which yields Without loss of generality we can also assume b k,j ∈ (n, +∞) for all k, j (it is enough to replace b k,j with b k,j + 1∕k ), hence a sequence of positive real numbers { k } ∞ k=1 has to exist such that lim k→∞ k = 0 and Now let Ω k,j denote the interior of R k,j . By applying Theorem 2.14 and Remark 2.15, we find a closed set F k,j ⊂ Ω k,j and an open set U k,j ⊂ Ω k,j satisfying and In particular, if define then we get Also one obviously has Observe that is a set of measure zero, by (13) and (14). Moreover, if then there exists l x ≥ 1 such that namely Hence, by (15) and (16) |d F k (x) − k (x)| < k for all k > l x .

Hence and by the first inclusion in (19) we obtain
Observe that {F l } ∞ l=1 is an increasing sequence of closed sets. The only thing left to prove is that each F l is uniformly n-dense. For this purpose, observe first that for all x ∈ F (n) l ∩ E⧵N we have by Definition 2.7, Proposition 4.3 and (17). Now the conclusion follows from the equivalence F (n) l ∩ E⧵N ∼ F l .