SUPERDENSITY WITH RESPECT TO A RADON MEASURE ON R n

. We introduce and investigate superdensity and the density degree of sets with respect to a Radon measure on R n . Some applications are provided. In particular we prove a result on the approximability of a set by closed subsets of small density degree and a generalization of Schwarz’s theorem on cross derivatives.


Introduction
Let us consider a Radon outer measure µ on R n and a µ measurable set E ⊂ R n .Then a celebrated result (cf.[17,Cor.2.14]) states that for µ almost all x ∈ E the set E is µ-dense at x, i.e., ) is verified, then we can pose the problem of defining a number d µ E (x) that exactly quantifies the density of E (w.r.t.µ) at x.A natural way (not the only way, certainly!) to solve this problem is as follows: • First we say that x is an h-superdensity point of E (w.r.t.µ) if h ∈ [0, +∞) and µ(Br(x)\E) µ(Br (x)) = o(r h ), as r → 0+; • Then we define the density degree of E (w.r.t.µ) at x, denoted by d µ E (x), as the supremum of all h ∈ [0, +∞) such that x is an h-superdensity point of E.
In our previous work we have obtained a number of results concerning superdensity with respect to the Lebesgue outer measure L n and the purpose of the present paper is to generalize some of these results.
In this introduction we want to summarize the most significant parts of the paper.Section 4 is devoted to prove some properties of the operator b µ,h : 2 R n → 2 R n (with h ∈ [0, +∞)) defined as follows b µ,h (A) := x ∈ spt µ lim sup Roughly speaking, b µ,h (A) is the set of all x ∈ spt µ such that the relative size of A in B r (x) is asymptotically larger than r h (as r → 0+).In Proposition 4.1 we find that b µ,h is a base operator, i.e., b µ,h (∅) = ∅ and for all A, B ∈ 2 R n .Moreover, if A µ,h denotes the set of all h-superdensity points of A (w.r.t.µ), then Hence b µ,h determines a topology τ b µ,h on R n which is finer than the ordinary Euclidean topology and such that A ∈ τ b µ,h if and only if A ∩ spt µ ⊂ A µ,h , cf.Proposition 4.2.There are two main results in this paper.The first one, Theorem 4.1, generalizes [7,Prop.3.2].It provides assumptions under which, in particular, the following property occurs (for any open set Ω ⊂ R n ): For every ε > 0 there exists an open set A ⊂ Ω such that µ(A) < ε and A is so "scattered" that the inclusion Ω ∩ spt µ ⊂ b µ,h (A) holds whenever h exceeds a certain value which does not depend on ε.Here is the full statement: Theorem 4.1.Let µ be non-trivial, i.e., spt µ = ∅.Suppose that there exist C, p, q, r ∈ (0, +∞) such that q ≤ min{n, p} and for all x ∈ spt µ and r ∈ (0, r).The following properties hold for all ε > 0 and h > np q − q (note that np q − q is non-negative): (1) If Ω ⊂ R n is a non-empty bounded open set, then there exists an open set A ⊂ Ω such that µ(A) < ε, In the special case when the set A can be chosen so that we have b µ,h (A) = Ω ∩ spt µ.
An example of application of Theorem 4.1 to the Radon measure carried by a regular surface in R n is given in Subsection 5.2.Another application is Proposition 6.3, which generalizes a property stated in [8,Prop.5.4].It provides a result on the approximability of a set by closed subsets of small density degree (w.r.t.µ): Proposition 6.3.Let µ be non-trivial and assume that: (i) There exist C, p, q, r ∈ (0, +∞) such that q ≤ min{n, p} and for all x ∈ spt µ and r ∈ (0, r); (ii) It is given a non-empty bounded open set Ω ⊂ R n with the following property: There exists an open bounded set for all h > m := np q − q.Then for all H ∈ 0, µ Ω there exists a closed subset F of Ω such that µ(F ) > H and The second main result generalizes the classical Schwarz theorem on cross derivatives (cf.Remark 7.1 below).Here is the statement: , a couple of integers p, q such that 1 ≤ p < q ≤ n and x ∈ R n .Assume that: (i) For i = p, q, the i-th distributional derivative of µ is a Borel real measure on R n also denoted D i µ, so that we have µ(Bρr (x)) (note that σ is decreasing); (iv) For i = p, q, one has lim r→0+ rµ(Br (x)) = 0 (where |D i µ| denotes the total variation of D i µ).
Among the results obtained in our previous work are several of the same kind as Theorem 7.1, in the special case µ = L n .They were then applied to describe the fine properties of sets of solutions of differential identities under assumptions of non-integrability.The simplest example that we can mention is then we conclude that A L 2 ,1 = ∅, regardless of f , even though there are functions f such that L 2 (A) > 0 (cf.[4,Theorem 2.1]).In particular, the density degree of A (w.r.t.L 2 ) is less than or equal to 1 everywhere and this gives us fairly accurate information about the fine structure of A. Similar arguments have been used, for example: • In [9], to prove that, given a C 1 smooth n-dimensional submanifold M of R n+m and a non-involutive C 1 distribution D of rank n on R n+m , the tangency set of M with respect to D can never be too dense.• In [10,11], to obtain results about low density of the set of solutions of the differential identity G(D)f = F , for certain classes of linear partial differential operators G(D), under assumptions of non-integrability on F .In connection with the results in [4] and [9], we would like to mention the paper [1] on the structure of tangent currents to smooth distributions.The application of superdensity used in [12] is a first successful attempt to extend the theory developed so far for the Lebesgue measure to other contexts (tangency of generalized surfaces as considered in [1]).At the same time, it gives us reason to believe that it is interesting to continue working on generalisation.It is in this sense that the present work, which provides a superdensity theory for Radon measures on R n , should be understood.

Basic notation and notions
2.1.Basic notation.The Lebesgue outer measure on R n and the s-dimensional Hausdorff outer measure on R n are denoted by L n and H s , respectively.The i-th partial derivative, either classical or distributional, will be denoted by D i .The ordinary topology of R n is denoted by is the open ball in R k , with centre x and radius r (k does not appear in the notation as its value will be made clear from the context).The family of all Radon outer measures on R n is denoted by R. If µ ∈ R then M µ is the σ-algebra of all µ measurable sets.When two subsets A and B of R n are equivalent with respect to µ ∈ R, i.e., µ(A \ B) = µ(B \ A) = 0, we write . In particular, this equality occurs whenever E ∈ M µ has finite measure and F is a "Borel envelope" of E (that is Let E be open.Then E ⊂ E µ,h and the inclusion can be strict, e.g., for µ = L n and Any base operator b is obviously monotone and determines a topology on X that is defined as follows Jϕ(y) := det[(Dϕ) t × (Dϕ)](y) 1/2 > 0 for all y ∈ G.We observe that H k ϕ(G) ∈ R.
Lemma 3.1.Let L be a real symmetric matrix of order k such that det L = 0 and (Lv) Proof of Proposition 3.1.Let us consider an arbitrary y ∈ G.We have to prove that namely, setting for simplicity µ := H k ϕ(G), To this end we observe that for all z ∈ R k .If • denotes the Hilbert-Schmidt norm of matrices and we define for all z ∈ R k .Hence, for all r > 0 and z ∈ ∂B r (y), we obtain where Observe that: provided r is small enough.Recalling also the area formula (cf.[14,Cor. 5.1.13]),it follows that this set of inequalities holds for r small enough: Hence, the statement (3.2) follows easily.
for all h ∈ [0, +∞) and A ⊂ R n .Hence and recalling (2) of Proposition 4.1 (or also simply by Definition 2.1), we obtain The same arguments used in [7, Prop.3.1]yield the following proposition.
Proposition 4.2.Let µ ∈ R and h ∈ [0, +∞).The following facts hold: The proof of Theorem 4.1 below is a non-trivial adaptation of the argument used to prove [7,Prop.3.2].We need to make a premise about lattices, which we include in the following remark.
Remark 4.1.We consider three positive integers R, β, k and set L k be the points of the lattice ) n and define the corresponding cells (which we will simply call k-cells) as Observe that the k-cells form a partition of [−R, R) n .Now let S be an infinite subset of [−R, R) n and denote by N k the number of k-cells intersecting S. Obviously one has N k ≤ N k+1 (for all k ≥ 1) and N k → +∞ (as k → +∞).Then we can easily find a countable family {P j } ⊂ S such that the following property holds, for all k ≥ 1: Each one of the k-cells intersecting S contains one and only one point of {P 1 , P 2 , . . ., P N k }.Under the assumptions above, we finally define Λ := ∪ +∞ k=1 Λ k and we say that {P j } is a Λ-distribution of S. Theorem 4.1.Let µ ∈ R be non-trivial, i.e., spt µ = ∅.Suppose that there exist C, p, q, r ∈ (0, +∞) such that q ≤ min{n, p} and for all x ∈ spt µ and r ∈ (0, r).The following properties hold for all ε > 0 and h > np q − q (note that np q − q is non-negative): In the special case when the set A can be chosen so that we have Proof.First of all observe that, by (2.1) and (4.1), we have µ(spt µ) > 0 and (4.5) µ({x}) = 0, for all x ∈ spt µ.
Hence spt µ is a non-countable set.That said, we can proceed to prove (1) and ( 2).
Proof of (1).If For if this were not true, x ∈ b µ,h (Ω) would exist for a certain h ∈ (0, +∞) and this would imply µ((B r (x)\{x})∩Ω) > 0 for all r > 0 (by (4.5)), which contradicts (4.6).Now, in the special case when (4.3) holds, the equality (4.7) yields ∂Ω ∩ spt µ = ∅ and it follows immediately from this that the second statement is also true.Thus, we can assume that Ω ∩ spt µ = ∅.This assumption and (2.2) (or (4.1)) imply that there exists an open ball B ⊂ Ω such that µ(B) > 0, hence From this fact and (4.5), it follows that Ω ∩ spt µ is a non-countable set.Now consider ε > 0 and h > np q − q.Define m := (h + q)q p . and observe that Moreover let R and β be positive integers such that For k = 1, 2, . .., we define and note that (4.10) ρ k < r by (4.9).Then, by recalling Remark 4.1 and the notation therein, we can find a Λdistribution {P j } ∞ j=1 of spt µ ∩ Ω.We set (for k = 1, 2, . ..) A k and observe that By (4.9), (4.10), (4.11) and assumption (4.1), we get Let us prove that To this end, consider x ∈ Ω ∩ spt µ and chose K x > 0 such that Obviously, for every k ≥ K x + 1 there exists a k-cell containing x.This k-cell must also contain a point of {P 1 , P 2 , . . ., P N k }, which we denote by Q k (cf.Remark 4.1).Observe that Then, for all k ≥ K x + 1 and y ∈ B ρ k (Q k ), we find (recalling (4.9) and (4.8) too) In particular From (4.13) and (4.14), recalling (4.10) and (4.1) too, we obtain C 1+p/q .Hence, by (4.1) and recalling the definition of m, we obtain (for k large enough) which shows that x ∈ b µ,h (A) and concludes the proof of (4.12).By recalling that • Ω is closed with respect to τ b µ,h (cf.(2) in Proposition 4.2), we can now complete the proof of (4.2):
Remark 4.2.Obviously, condition (4.1)only makes sense if q ≤ p.Moreover, if q > n this condition implies that spt µ is empty.In fact, if we assume spt µ = ∅ (and q > n), then we obtain the following contradiction: • On the one hand, as observed at the beginning of the proof of Theorem 4.1, one would have µ(spt µ) > 0; • On the other hand, by [17, Th.6.9],we have µ(spt µ) = 0.These considerations make it clear why we assumed q ≤ min{n, p} in Theorem 4.1.
Remark 4.3.Let p, q be as in Theorem 4.1.Then it is easy to verify that np q − q = 0 if and only if p = q = n.Remark 4.4.We observe that: ( where T x S and T x (∂Ω)) are the tangent space of S at x and the tangent space of ∂Ω at x, respectively.We observe that then we also have dim(T x S ∩ T x (∂Ω)) = k − 1 and this fact implies that near x the set ∂Ω ∩ S is an imbedded Then, with a standard argument based on the area formula, we can prove that x ∈ b µ,0 (Ω) (hence x ∈ b µ,h (Ω) for all h ∈ [0, +∞)).Therefore, if we now assume that S and ∂Ω meet transversely everywhere (i.e., at every point in ∂Ω ∩ S), then we find ∂Ω ∩ S ⊂ b µ,0 (Ω).This does not imply that condition (4.3) is verified.For example, consider the case n := 3, k := 2 and In this case S and ∂Ω meet transversely everywhere and ∂Ω ∩ S = b µ,h (Ω) for all h ∈ [0, +∞).Hence we have also (0, 0, 1) ∈ b µ,h (Ω) Remark 4.6.It is natural to ask whether Theorem 4.1 can be extended to the case that r depends on x ∈ spt µ.After trying to prove such a generalisation, we are inclined to believe that the answer is negative, but we have no counterexamples.Observe that m 00 > 0 by Lemma 3.1.Furthermore, since Dϕ is continuous, we easily see that there must exist r 0 ∈ (0, 1] such that σ r (y) := max for all y ∈ G, z ∈ ∂B r (y) and r ∈ (0, 1], where m 1 is defined as in (3.4).From (5.2) and (5.3) it follows that for all y ∈ G and r ∈ (0, r 0 ].Now, using (5.4) , we can proceed to the proof of (5.1): • We first prove by contradiction the following claim: There exist C 1 , r 1 ∈ (0, +∞) such that for all x ∈ spt µ = ϕ(G) and r ∈ (0, r 1 ].If this were not true, for each positive integer j there would exist y j ∈ G and ρ j ∈ (0, 1/j] such that (5.6) Since G is compact we can assume that y j → ȳ ∈ G, as j → +∞.On the other hand, by the second inclusion in (5.4) and the area formula, we have which contradicts (5.6).Thus the claim above has to be true.
• From the first inclusion in (5.4) and the area formula it follows that Jϕ dL k for all y ∈ G and r ∈ (0, m 00 r 0 ].Thus, since Jϕ is bounded in G, there must exist a positive constant C 2 (which does not depend on x and r) such that for all x ∈ spt µ = ϕ(G) and r ∈ (0, m 00 r 0 ].
In the special case when the set A can be chosen so that we have ( There is an open set U ⊂ R n satisfying
First of all observe that if E ⊂ R n and x ∈ R n , then the set {h ∈ [0, +∞) | x ∈ E µ,h } is a (possibly empty) interval.
Definition 6.1.Let E be a subset of R n .Then the density degree of E (w.r.t.µ) is the function d µ E : R n → {−n} ∪ [0, +∞] defined as follows: For m ∈ [0, +∞] we also define When the following identity holds Remark 6.1.The following trivial facts occur: (1) for all m ∈ [0, +∞).Observe that the strict inclusion can occur, e.g., for µ := L n and E := B r \ {0} (in such a case one has int µ,m E = B r ).
This proposition collects some very simple (nevertheless interesting) facts.
• If we assume that the first claim is true, then, by recalling also (3), we obtain This proves the first formula in the second claim.It also proves that cl µ,m E = µ cl µ,m E\ int µ,m , hence the last formula in the second claim follows by recalling (5).
Proposition 6.2.Let E be a measurable subset of R n .Then the set Now we prove a result about approximation of a set, given as the closure of an open set, by closed subsets having small density degree (w.r.t.µ).The proof is obtained by adapting the argument used in [8,Prop.5.4].
Proposition 6.3.Assume that: (i) There exist C, p, q, r ∈ (0, +∞) such that q ≤ min{n, p} and for all x ∈ spt µ and r ∈ (0, r); (ii) It is given a non-empty bounded open set Ω ⊂ R n with the following property: There exists an open bounded set for all h > m := np q − q.Then for all H ∈ 0, µ Ω there exists a closed subset F of Ω such that µ(F ) > H, int µ,m F = µ ∅.
Proof.Let j be an arbitrary positive integer.Then, by Theorem 4.1, there exists an open set A j ⊂ Ω ′ such that (6.1) Define Then K is closed and by (6.1).Moreover, by ( 2), (3), (5) of Proposition 4.1 and (6.1), we have for all j.Moreover, for each k ∈ (m, +∞) we can find j such that k > h j , hence of Remark 2.1.Recalling (2) of Proposition 6.1, we obtain

Now define
Then F is a closed subset of Ω and (again by (2) of Proposition 6.1) 2), where Hence µ(F ) > H. Then for all H ∈ 0, H k A there exists a closed set E ⊂ A such that where λ := H k ϕ(G).In particular, E is a uniformly (λ, 0)-dense set.
Proof.Let us consider the bounded open set and observe that, by (6.3), we have also where From Proposition 6.3 (with n = p = q = k and µ = L k ) and recalling (1) of Remark 4.4, it follows that a closed set K ⊂ D has to exist such that (6.7) Then consider h ∈ [0, +∞) and the closed set Observe that (6.8) by (6.5) and ( 7) in Remark 2.1.Hence and by the area formula (cf.[14,Cor. 5.1.13])we obtain The inequality in (6.4) now follows easily from (6.6), (6.7) and (6.9).
Remark 6.5.In general, Proposition 6.3 does not provide the optimal result.For example, if we apply Proposition 6.3 directly to the measure λ carried by a k-dimensional imbedded C 1 submanifold of R n with C 1 boundary we get a worse result than that obtained in Corollary 6.1.To verify this fact, let us consider G and ϕ as in Section 3 and further assume that ∂G is of class C 1 .We observe that hypothesis (i) of Proposition 6.3 is verified, with where E := F ∩ϕ(G), which is closed with respect to the topology induced in ϕ(G) by τ (R n ).Therefore, this argument does not prove the result obtained in Corollary 6.1, namely, that there are closed subsets of A of arbitrarily close measure to H k A that are also uniformly (λ, 0)-dense.

A Schwarz-type result
We will prove the following result that generalizes the classical Schwarz theorem on cross derivatives (cf.Remark 7.1 below).
Theorem 7.1.Let us consider µ ∈ R, an open set Ω ⊂ R n , f, G, H ∈ C 1 (Ω), a couple of integers p, q such that 1 ≤ p < q ≤ n and x ∈ R n .Assume that: (i) For i = p, q, the i-th distributional derivative of µ is a Borel real measure on R n also denoted D i µ (with no risk of misinterpretation), so that we have µ(Bρr (x)) (note that σ is decreasing); (iv) For i = p, q, one has lim r→0+ Proof.Let ρ ∈ (0, 1) and consider g ∈ C 2 c (B 1 (0)) such that 0 ≤ g ≤ 1, g| Bρ(0) ≡ 1 and For every real number r such that 0 < r < dist(x, R n \ Ω), we define g r ∈ C 2 c (B r (x)) as g r (y) := g y − x r , y ∈ R n and observe that (for all y ∈ B r (x) and i = 1, . . ., n) Then, after a simple computation in which we use only (i), the definition of A in (ii) and the identity D p D q g r = D q D p g r , we arrive at the following equality (where B r and B ρr stand for B r (x) and B ρr (x), respectively): Hence, by also recalling the polar decomposition theorem (cf.[2, Cor.1.29])and (7.1), we obtain where C is a suitable positive constant independent from r and ρ.Consequently, C can be chosen such that we have for all r, ρ ∈ (0, 1).Hence, by assumptions (iii) and (iv), we obtain for every ρ in a left neighborhood of 1.The conclusion follows from assumption (iii).) cos(jx 1 ) sin(jx 2 ).From (7.4) and the equality (D 1 ϕ j )(t, t) = 2tη ′ (2t 2 ) cos(jt) sin(jt) − jη(2t 2 ) sin 2 (jt) Remark 7.3.If in Corollary 7.1 we take h ≡ 1 then assumptions (i) and (iii) are trivially verified at every x ∈ R n .Recalling also (1) of Remark 2.1, we conclude that D p H = D q G in Ω ∩ A (n+1) .In particular, the following property immediately follows: If f ∈ C 1 (Ω), Moreover, by the divergence theorem, we have Thus assumption (i) of Theorem 7.1 is trivially verified, while (ii) yields assumption (iv) of Theorem 7.1.

Declarations
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r (x) ∩ E) µ(B r (x)) = 1, which is equivalent to lim r→0+ µ(B r (x) \ E) µ(B r (x)) = 0, where B r (x) denotes the open ball in R n , with centre x and radius r.If the condition (1.1 out that τ b is the finest topology τ on X such that, for all A ⊂ X, the closure of A w.r.t.τ contains b(A).If X = R n and b(A) denotes the ordinary closure of A ⊂ R n , then b is a base operator and τ b = τ (R n ).

3 .
Superdensity w.r.t. the measure carried by a regular surface Let G be a bounded open subset of R k and consider

Corollary 5 . 1 .( 1 )
7) yield (5.1) with C := max{C 1 , C 2 } and r := min{r 1 , m 00 r 0 }.Now, by applying Theorem 4.1 with µ = H k ϕ(G) and p = q = k (taking Proposition 5.1 into account), we obtain: The following properties hold for all ε > 0 and h > n − k: If Ω ⊂ R n is a bounded open set, then there exists an open set A ⊂ Ω such that

Remark 6 . 3 .
Proposition 6.1 holds whatever negative value is assigned, in Definition 6.1, to the restriction of d µ E to E µ,0 .We chose −n only because this way the function n + d L n E coincides with the density degree function d E defined in [8, Def.5.1].

Proposition 5 . 1 .
Now let A ⊂ ϕ(G) be open with respect to the topology induced in ϕ(G) by τ (R n ) and assume that (6.3) holds.By a standard argument, it follows that an open setΩ ⊂ R n exists such that A = Ω ∩ ϕ(G), A = Ω ∩ ϕ(G).Since spt µ is bounded, there is an open ball B ⊂ R n such that Ω ⊂ B and ∂B ∩ spt µ = ∅.Hence (ii) of Proposition 6.3 is trivially verified, with Ω ′ = B. Now consider any H ∈ 0, H k A and observe that H k A = λ Ω .Then, by Proposition 6.3, there exists a closed subset F of Ω such that λ(F ) > H and int λ,n−
Def.1.12].The total variation of a Borel real measure λ on R n is denoted by |λ| Let µ ∈ R, h ∈ [0, +∞) and E, F ⊂ R n .Then it can easily be verified that the following properties hold true:(1) If µ = L n then the set of all h-superdensity points of E w.r.t.µ coincides with the set of all (n + h)-density points of E, i.e., E L n 1) If µ = L n then condition (4.3) is verified whenever ∂Ω is Lipschitz (for all h ∈ [0, +∞)).Hence Theorem 4.1 yields immediately [7, Prop.3.2].(2) No regularity assumption on ∂Ω will suffice to ensure that condition (4.3) is verified for all µ ∈ R. For example, if Ω is a ball and µ := H n−1 ∂Ω then ∂Ω ∩ spt µ = ∂Ω and b µ,h (Ω) = ∅ (for all h ∈ [0, +∞)).Remark 4.5.Let µ := H k S, where S is an open imbedded k-submanifold of R n of class C 1 with k ≤ n − 1. Moreover let ∂Ω be of class C 1 and assume that S and ∂Ω meet transversely at x, namely