General Hausdorff functions, and the notion of one-sided measure and dimension

c 2008 by Institut Mittag-Leffler. All rights reserved

General Hausdorff functions, and the notion of one-sided measure and dimension

Claude Tricot

Abstract. The main facts about Hausdorff and packing measures and dimensions of a Borel setE are revisited, usingdetermining set functionsφα:BE!(0,∞), whereBE is the family of all balls centred onE andαis a real parameter. With mild assumptions onφα, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension.

Given a bounded open setV inR^D, these notions are used to introduce theinteriorandexterior measures and dimensions of any Borel subset of∂V. We stress that these dimensions depend on the choice ofφα. Two determining functions are considered, φα(B)=Vol_D(B∩V) diam(B)^α−D andφα(B)=Vol_D(B∩V)^α/D, where Vol_D denotes theD-dimensional volume.

1. Introduction

Hausdorff and packing measures were first intended to generalize the notions of length, area, volume etc. In his 1919 paper [4], F. Hausdorff uses generaldetermin- ing functions (also calledHausdorff,gauge orconstituent functions)φ: R⁺!R⁺to define the Hausdorff measure in a metric space as

H^φ(E) = lim

ε!0inf

i≥0

φ(diam(E_i)) :E⊂

i≥0

E_i and diam(E_i)≤ε

. (1)

Let id be the identity function, so that id^α(t)=t^αfor allt∈Randα>0. The scale of functions Φ={id^α:α>0}is used to obtain a large family of Hausdorff measures H^idα, rather denoted by H^α for simplicity. Up to some constants, H¹ gives the proper definition for the notion of length, H²for the notion of area etc. For many references concerning the classical theory, see [2].

The Hausdorff dimension of a set E is introduced as a critical exponent:

dim(E)=inf{α:H^α(E)=0}. The dimension is a real number in this context, but a different choice of Φ could lead to a different notion of dimension, as pointed out

by Hausdorff himself in his seminal paper. In general the dimension is a Dedekind cut in a scale of functions.

The value of H^α(E) is 0 whenα>dim(E), +∞whenα<dim(E). When H^α(E) is non-zero and finite, its exact value is known only for very particular sets like curves of finite length or symmetric Cantor sets. As shown by A. S. Besicovitch, a finite measure μsuch thatμ(E)>0 may help to get estimations of H^α(E), by using the μ-densities μ(B(x, ε))/ε^α, whereB(x, ε) denotes the ball of centrexand radiusε.

A typical result [6] stands as follows: If aandbare such that a≤lim sup

ε!0

μ(B(x, ε)) (2ε)^α ≤b

for all x∈E, then a≤H^α(E)≤2b. A better result may be obtained by changing somewhat the definition of H^α. If we consider only covers of E by centred balls, thecentred covering measure c^α(E) is obtained through a two-stages operation [9].

Then, with the same assumptions, the inequalitya≤c^α(E)≤b is always true.

The corresponding result for the dimension is often attributed to P. Billings- ley [1]: Let μbe a finite measure such that μ(E)>0. If there exists αandβ such that

α≤lim inf

ε!0

logμ(B(x, ε)) logε ≤β for allx∈E, then dim(E)∈[α, β].

Giving a sense to the above formulas after exchanging lim inf and lim sup is possible, at the condition of introducing new notions of measure and dimension.

Using the scale Φ={id^α:α>0}, this was done in [12] for the dimension and in [11]

and [9] for the measures, by introducing the packing measures and dimension. In the present paper we will give a detailed account of these notions, in a more general framework.

The above density results have been extended to functions φ: (0,∞)!(0,∞) which are increasing, continuous at 0, and such that φ(2ε)/φ(ε) is bounded [9].

More generally, φ(diam(E_i)) may be replaced by φ(E_i), where φ is a specific set function: See for example [7], where a multifractal analysis of measures usesφ(E_i)=

diam(E_i)^tν(E_i)^q for some prescribed measureν. In [8], φis a function defined on balls centred on the set E under analysis. This is also our point of view in the present paper.

LetD≥1 and Vol_D be theD-dimensional volume inR^D. For any Borel setE, let B_E be the family of closed balls centred on E. In Section 2 we use functions φ: B_E!(0,∞) to define covering and packing Borel measures H^φ and p^φ. We get corresponding results involving the μ-densities and prove the usual inequality

H^φ≤p^φ. For any Borel set E, replacing the measureμ by the restriction to E of the measures H^φand p^φ allows us to statelocal density results in Section 3, giving rise to a notion of rectifiability. Until this point, there is no assumption whatsoever made on the determining functionφ. We introduce parameterized scales of functions Φ={φ_α:α∈R}in Section 4 to define the related covering and packing dimensions, dimΦ and DimΦ. Section 5 deals with the corresponding box dimension. With the help of mild assumptions on functions φ_α, we introduce a pre-measure and show that the corresponding critical index Δ_Φ verifies some of the fundamental properties of the usual box dimension, for example ΔΦ(E)=ΔΦ(E). We investigate the particular caseφ_α(B)=ν(B) diam(B)^α−D, where ν is a finite, Borel measure.

IfE(ε) is the Minkowski sausage (set of points at a distance≤εfromE), we show that Δ_Φ(E)=lim sup_ε!∞(D−logν(E(ε)))/logε.

Section 6 deals with the main purpose of this paper: defining lateral or one- sided measures and dimensions on both sides of a closed curve. This has already been done for the box-counting dimension [15] but for the Hausdorff dimension these notions seem to be new. More generally, being given an open set V and E⊂∂V, we perform a fractal analysis of E relative to V, by using determining functions φ: B_E!(0,∞) such that φ(B) depends on B∩V. There are several ways to do this. We pay special attention to the function

φ_α(B) =Vol_D(B∩V)

Vol_D(B) diam(B)^α.

It involves a notion of covering and packing interior measures (from the results of Section 2), and interior dimensions (from Section 4) denoted by dim_int and Dim_int. Being given a finite measure onEsuch thatμ(E)>0, we can establish relationships between those dimensions and the quantities

logμ(B)

log(diam(B))+D−log Vol_D(B∩V) log(diam(B)) . It is worth pointing out that the following characteristic quantity

−D+log Vol_D(B∩V) log(diam(B))

has already been considered in [5] to perform a multifractal analysis of ∂V, seen from the interior ofV.

Considering the interior of the complement ofV, it is also possible to define exterior dimensionsdimextand Dimext. Relationships between these dimension are obtained, namely: If Vol_D(∂V)=0 andE⊂∂V, then

max{dimint(E),dimext(E)} ≤dim(E)≤Dim(E) = max{Dimint(E),Dimext(E)}. (2)

Some examples are studied. An interesting feature of this determining functionφ is that the interior dimension of a singleton depends on the shape of∂V, and may take all values between −∞ and 0. We also construct an open set V in R, then inR², such that the first inequality in (2) is strict. Finally we point out in Section 7 that this is not the only approach to a one-sided fractal analysis: other determining functions might be used, giving other results.

2. φ-measures

Being given a set E in R^D, a net over E is a family F of sets such that every point inE belongs to a sequenceE_n ofF such that diam(E_n)!0. In other words, F is a net if, for everyε>0, one can extract from F a cover of E by sets of diameter ≤ε. It is sometimes called a Vitali covering, or a fine covering. The family of all closed centred balls BE is a net.

Acentred net overE will be a subfamilyF ofBE such that everyx∈E is the centre of a sequence of balls in F whose diameters tend to 0.

For every set familyF,

F denotes the union of the sets inF, as usual.

We will make use of theBesicovitch lemma, in the two following forms:

Lemma 2.1. Let E be a bounded set, andF ⊂BE be such that everyx∈E is the centre of some ball in F. FromF one can extract a numberK≤6^D of families R1, ...,R_K such that_K

n=1R_n covers E and every R_n consists of disjoint centred balls.

This result implies the following consequence:

Lemma 2.2. LetE be a Borel set andμbe a Borel measure such thatμ(E)<

+∞. From every centred net F over E we can extract a family R, consisting of disjoint balls, such that μ(E\

R)=0.

Let us recall that a set functionF issubadditiveifF(E1∪E2)≤F(E1)+F(E2), andσ-subadditive if for any sequence{E_n}n≥0, F(

n≥0E_n)≤

n≥0F(E_n).

Letφbe an application defined onB_E, with values in (0,+∞).

2.1. φ-covering measures

For everyε>0, let C^φ_ε(E) = inf

i≥0

φ(B_i) :B_i∈ B_E, E⊂

i≥0

B_iand diam(B_i)≤ε

. (3)

It is a decreasing function of ε, with values in [0,+∞]. As a set function it is σ-subadditive. The quantity

C^φ(E) = lim

ε!0C^φ_ε(E) (4)

is defined like the Hausdorff measure. As a set function, C^φ isσ-subadditive, but not increasing. Hence it is not a measure. To change it into an increasing function, it is necessary to perform the following operation:

c^φ(E) = sup{C^φ(F) :F is Borel andF⊂E}. (5)

This new set function is increasing and σ-subadditive. Moreover we can set c^φ(∅)=0. Then c^φ is an outer measure. Since it is also metric, c^φ is a Borel measure, called the centred φ-covering measure [9]. The next two results will be helpful to establish a density theorem for c^φ.

Proposition 2.3. Let μ be a Borel measure and F be a centred net over E.

Let us assume that

φ(B)≤μ(B) for allB∈ F. Then c^φ(E)≤μ(E).

Proof. Suppose that μ(E)<∞. Let ε>0, and for every open V, G(V, ε)=

{B∈F:diam(B)≤εandB⊂V}.

(a) SinceEis Borel, there exists an open setV_Esuch thatE⊂V_Eandμ(V_E)≤ μ(E)+ε. The familyG(V_E, ε) is a centred net overE. Using Lemma 2.2, we can ex- tract fromG(V_E, ε) a sequence{B_i}i≥0of disjoint balls such thatμ(E\

i≥0B_i)=0.

Therefore

i≥0

φ(B_i)≤μ(V_E)≤μ(E)+ε.

(b) Let F be a Borel set inE. There exists an open setV_F such thatF⊂V_F andμ(V_F)≤μ(F)+ε. Using Lemma 2.1, one can extract fromG(V_F, ε) a numberK of familiesR1, ...,R_Ksuch thatR_nis made up with disjoint balls andF⊂_K

n=1R_n. For everyn,

{φ(B) :B∈ R_n} ≤μ(V_F)≤μ(F)+ε,

so that

{φ(B):B∈_K

n=1Rn}≤K(μ(F)+ε). Applying this toF=E\

i≥0B_i, we get a cover_K

n=1R_n ofE\

i≥0B_i such that φ(B) :B∈

K n=1

Rn

≤Kε.

(c) Since{B_i}i≥0∪(_K

n=1Rn) is anε-covering ofE, C^φ_ε(E)≤

i≥0

φ(B_i)+

φ(B) :B∈^K

n=1

R_n

≤μ(E)+(K+1)ε.

Lettingε!0, we get C^φ(E)≤μ(E). This result is true for any Borel subsetGofE:

C^φ(G)≤μ(G)≤μ(E).

Taking the supremum over allG⊂Egives c^φ(E)≤μ(E).

Here is a result in the other direction:

Proposition 2.4. Let ε₀>0,Fε0 be the family of all balls centred onE such that diam(B)≤ε0, andμ be a Borel measure such that

φ(B)≥μ(B) for all B∈ F_ε0. Then c^φ(E)≥μ(E).

Proof. Letε∈(0, ε0]. For every cover{B_i}_i≥0 ofEby balls centred onE such that diam(B)≤ε,

i≥0

φ(B_i)≥

i≥0

μ(B_i)≥μ(E),

so that C^φ_ε(E)≥μ(E). Letting ε!0, shows that C^φ(E)≥μ(E). Since c^φ≥C^φ we get the desired result.

Now we can set up the main density result. It allows us to give estimates of c^φ(E) with the help of theμ-densities μ(B)/φ(B).

The usual conventions hold: for every a>0, 0/a=0 anda/0=+∞. The cases of indetermination are ⁰₀ and +∞/+∞.

Theorem 2.5. Let E be a Borel set and μbe a Borel measure. Then

x∈Einf lim sup

ε!0

μ(B(x, ε))

φ(B(x, ε))≤ μ(E) c^φ(E)≤sup

x∈Elim sup

ε!0

μ(B(x, ε)) φ(B(x, ε)), (6)

except in the cases of indetermination.

Proof. (a) Let D₁ be the left-hand side of (6). Assume that D₁>0. Let 0<

A1<D1, andF={B∈BE:A1φ(B)≤μ(B)}. The family F is a centred net overE.

Let μ^∗=(1/A1)μ. Sinceφ(B)≤μ^∗(B) for all B∈F, Proposition 2.3 gives c^φ(E)≤ μ^∗(E), so thatA₁c^φ(E)≤μ(E). LettingA₁tend toD₁givesD₁c^φ(E)≤μ(E). This may be written asD1≤μ(E)/c^φ(E), except in the indetermination cases.

(b) Let D2 be the right-hand side of (6). Assume thatD2<∞. LetA2>D2, and

E_k=

x∈E:ε≤1

k⇒μ(B(x, ε))≤A₂φ(B(x, ε))

, so thatE=

k≥1E_k. Proposition 2.4 implies thatA2c^φ(E_k)≥μ(E_k). Sinceμ and c^φ are Borel measures andE_k is an increasing sequence of sets,A2c^φ(E)≥μ(E). To conclude, letA₂ tend toD₂.

2.2. φ-packing measures

LetE be a bounded Borel set. For everyε>0,{B_i}_i≥0is anε-packing ofE if B_i∈BE, theB_i are disjoint, and diam(B_i)≤ε. Let

P^φ_ε(E) = sup

i≥0

φ(B_i) : {B_i}i≥0 is anε-packing ofE

. (7)

This set function is increasing and subadditive. As a function ofεit increases, and we can set the followingpre-measure

P^φ(E) = lim

ε!0P^φ_ε(E).

(8)

It is also increasing and subadditive, but notσ-subadditive. We get theφ-packing measurethrough the standard operation [12], [10],

p^φ(E) = inf

n≥0

P^φ(E_n) :E_n are Borel sets withE=

n≥0

E_n

. (9)

In particular, p^φ≤P^φ. Like H^φand c^φ, p^φis a metric, outer measure, and therefore a Borel measure. The main results relating the value of p^φ(E) to μ(E), for some measure μ, are very similar to those already proved for c^φ. We begin with two propositions and get a density theorem.

In the following proofs, it is necessary to assume thatE is bounded since the pre-measure P^φis used; but all results, as well as (9), may be extended to unbounded setsE without change.

Proposition 2.6. Letε₀>0,F_ε0 be the family of all balls centred on E such that diam(B)≤ε0, andμbe a Borel measure such that

φ(B)≤μ(B) for allB∈ F_ε0. Then p^φ(E)≤μ(E).

Proof. Assume thatμ(E)<∞. LetV be an open set such thatE⊂V. Let N be an integer such thatN≥1/ε0. For everyn≥N, let

E_n=

x∈E:B

x,1 n ⊂V

.

The sequence {E_n}_n≥N is increasing. If ε≤1/n and x∈E_n, then φ(B(x, ε))≤ μ(B(x, ε)). For everyε-packing{B_i}i≥0 ofE_n, we get

i≥0

φ(B_i)≤

i≥0

μ(B_i)≤μ(V).

Therefore P^φ_ε(E_n)≤μ(V). Lettingε!0 gives that P^φ(E_n)≤μ(V). We deduce that p^φ(E_n)≤μ(V) for alln≥N, and sinceE=

n≥0E_n, p^φ(E)≤μ(V). SinceEis Borel, we can chooseV such thatμ(V) is as close toμ(E) as we wish. Therefore p^φ(E)≤ μ(E).

Proposition 2.7. Let μ be a Borel measure and F be a centred net over E.

Let us assume that

μ(B)≤φ(B) for allB∈ F. Then p^φ(E)≥μ(E).

Proof. Letε>0, andGε={B∈F:diam(B)≤ε}. Using Lemma 2.2, fromGεwe can extract a disjoint family {B_i}i≥0 such thatμ(E)=μ(E∩(

i≥0B_i)). Therefore μ(E)=

i≥0μ(B_i)≤

i≥0φ(B_i)≤P^φ_ε(E). Letting ε!0 gives thatμ(E)≤P^φ(E).

This inequality is also true for any Borel subset of E. IfE=

k≥0E_k, then μ(E)≤

k≥0μ(E_k)≤

k≥0P^φ(E_k). Taking the infimum over all Borel covers ofE givesμ(E)≤p^φ(E).

Propositions 2.6 and 2.7 imply the following density theorem, symmetrical to Theorem 2.5:

Theorem 2.8. Let E be a Borel set,μbe a Borel measure. Then

x∈Einf lim inf

ε!0

μ(B(x, ε))

φ(B(x, ε))≤ μ(E) p^φ(E)≤sup

x∈Elim inf

ε!0

μ(B(x, ε)) φ(B(x, ε)), (10)

except in the cases of indetermination.

Proof. (a) Let D1 be the left-hand side of (10). Assume that D1>0. Let 0<A₁<D₁, and

E_k=

x∈E:ε≤1

k⇒A1φ(B(x, ε))≤μ(B(x, ε))

.

Proposition 2.6 implies thatA1p^φ(E_k)≤μ(E_k). Since E=

k≥1E_k and E_k is in- creasing,A1p^φ(E)≤μ(E). Then letA1 tend toD1.

(b) LetD2 be the right-hand side of (10). Assume thatD2<∞. LetA2>D2, and F={B∈B_E:μ(B)≤A2φ(B)}. The family F is a centred net over E. Prop- osition 2.7 givesμ(E)≤A₂p^φ(E). Then letA₂tend to D₂.

The relationships between the measures c^φ and p^φ are as usual:

Proposition 2.9. For any Borel set E and φ: B_E!(0,∞), the inequality c^φ(E)≤p^φ(E)holds.

Proof. Letε>0, andFε={B∈BE:diam(B)≤ε}. Since c^φ is a Borel measure, we may use Lemma 2.2 to extract a disjoint family {B_i}_i≥0 from F_ε such that c^φ(E\

i≥0B_i)=0. The set function C^φ_ε is subadditive, so that C^φ_ε(E)≤C^φ_ε

E∩

i≥0

B_i +C^φ_ε

E\

i≥0

B_i .

From the general inequalities C^φ_ε≤C^φ≤c^φ, we deduce that C^φ_ε(E\

i≥0B_i)=0. And C^φ_ε(E∩(

i≥0B_i))≤

i≥0φ(B_i)≤P^φ_ε(E), so that C^φ_ε(E)≤P^φ_ε(E).

Lettingε!0 gives that C^φ(E)≤P^φ(E).

For any F⊂E we get C^φ(F)≤P^φ(F)≤P^φ(E), so that c^φ(E)≤P^φ(E). If E=

n≥0E_n, then c^φ(E)≤

n≥0c^φ(E_n)≤

n≥0P^φ(E_n). Therefore c^φ(E)≤p^φ(E).

3. Local density inequalities

Local density inequalities are direct applications of the density theorems, where the measureμis replaced by the Borel measures c^φ or p^φ restricted toE. Without looking for exhaustiveness we present here a few basic results. Again, our statements do not require any assumption on the determining functionφ:BE!(0,∞).

LetE be a Borel set inR^D. We make use of the classical notations d^φ_c(x, E) = lim inf

ε!0

c^φ(B(x, ε)∩E)

φ(B(x, ε)) , d_c^φ(x, E) = lim sup

ε!0

c^φ(B(x, ε)∩E) φ(B(x, ε)) , d^φ_p(x, E) = lim inf

ε!0

p^φ(B(x, ε)∩E)

φ(B(x, ε)) , d_p^φ(x, E) = lim sup

ε!0

p^φ(B(x, ε)∩E) φ(B(x, ε)) .

Lemma 3.1. Suppose thatc^φ(E)<∞. LetF be a Borel subset ofE.

If c^φ(F)>0, then

x∈Finf d^φ_c(x, E)≤1≤sup

x∈Fd_c^φ(x, E).

(11)

If p^φ(F)>0, then

x∈Finf d^φ_c(x, E)≤c^φ(F) p^φ(F)≤sup

x∈Fd^φ_c(x, E).

(12)

Proof. For any Borel setG, let μ(G)=c^φ(G∩E). Thenμ is a Borel measure, such that

d_c^φ(x, E) = lim sup

ε!0

μ(B(x, ε)) φ(B(x, ε)),

and μ(F)=c^φ(F). A direct application of (6) gives (11). The condition c^φ(F)>0 ensures that the indetermination cases are avoided. Also,

d^φ_c(x, E) = lim inf

ε!0

μ(B(x, ε)) φ(B(x, ε)), and a direct application of (10) gives (12).

Equations (6) and (10) may also be used to obtain the symmetric results:

Lemma 3.2. Suppose thatp^φ(E)<∞. LetF be a Borel subset ofE such that p^φ(F)>0. Then

x∈Finf d^φ_p(x, E)≤p^φ(F) c^φ(F)≤sup

x∈Fd_p^φ(x, E), (13)

x∈Finf d^φ_p(x, E)≤1≤sup

x∈Fd^φ_p(x, E).

(14)

Corollary 3.3. If c^φ(E)<∞, thend_c^φ(x, E)=1 c^φ-almost everywhere, that is, for all x∈E except on a subset F such that c^φ(F)=0.

Proof. Let F_k=

x∈E:d^φ_c(x, E)≤1−1 k

and G_k=

x∈E:d_c^φ(x, E)≥1+1 k

. If c^φ(F_k)>0, then (11) implies that 1≤1−1/k. This is impossible, so that c^φ(F_k)=0.

If c^φ(G_k)>0, then (11) implies that 1≥1+1/k. This is impossible, so that c^φ(G_k)=0.

Taking countable unions ofF_k andG_k, we deduce that

c^φ({x∈E:d_c^φ(x, E)<1}) = c^φ({x∈E:d^φ_c(x, E)>1}) = 0.

Similar arguments and (14) are used for the following consequence:

Corollary 3.4. If p^φ(E)<∞, thend^φ_p(x, E)=1 p^φ-almost everywhere.

The following results deal with the notion of regularity:

Corollary 3.5. If p^φ(E)<∞, then the following are equivalent:

(i) c^φ(E)=p^φ(E);

(ii) d^φ_c(x, E)=d_c^φ(x, E)=1 p^φ-a.e.;

(iii) d^φ_p(x, E)=d_p^φ(x, E)=1 p^φ-a.e.

Sets that fulfill these conditions may be calledstronglyφ-regular.

Proof. (i)⇒(ii) and (iii) For any Borel set F⊂E, c^φ(F)≤p^φ(F). If c^φ(E)=

p^φ(E), then c^φ(F)=p^φ(F). This implies that c^φ-a.e. is equivalent to p^φ-a.e. in this case.

Let F_k={x∈E:d^φ_c(x, E)≤1−1/k}. If p^φ(F_k)>0, then (12) implies that 1≤ 1−1/k. This is impossible, so that p^φ(F_k)=0 for allk, or in other wordsd^φ_c(x, E)≥1 p^φ-a.e.

Similarly, define G_k={x∈E:d_p^φ(x, E)≥1+1/k} and use (13) to show that p^φ(G_k)=0 for allk, or in other wordsd_p^φ(x, E)≤1 p^φ-a.e.

With Corollaries 3.3 and 3.4, this suffices to prove (ii) and (iii).

(ii)⇒(i) LetF={x∈E:d^φ_c(x, E)≥1}. Equation (12) implies that c^φ(F)≥p^φ(F), hence c^φ(F)=p^φ(F). By hypothesis, p^φ(F)=p^φ(E), so that p^φ(E\F)=0 and c^φ(E\F)=0. Therefore c^φ(F)=c^φ(E). We conclude that c^φ(E)=p^φ(E).

(iii)⇒(i) This is shown by similar arguments, using (13).

Let us give an application, among many others, which deals with the Cartesian product of sets:

Proposition 3.6. LetD1≥1andD2≥1. We assume that R^D1+D2 is endowed with a metric which is the Cartesian product of the metrics in R^D1 and R^D2, so that for allε>0,x∈R^D1 andy∈R^D2, we have:

B((x, y), ε) =B(x, ε)×B(y, ε).

Let E be a Borel set of R^D1, F be a Borel set of R^D2, φ: B_E!(0,+∞) and ψ:B_F!(0,+∞).

If c^φ(E)<+∞andc^ψ(F)<+∞, then

c^φ(E)c^ψ(F)≤c^φ×ψ(E×F).

(15)

Notice thatφ×ψis defined onBE×F, and that

(φ×ψ)(B((x, y), ε)) =φ(B(x, ε))ψ(B(y, ε)).

Proof. Without loss of generality, we assume that c^φ(E)>0 and c^ψ(F)>0. For all Borel setsG₁⊂R^D1 andG₂⊂R^D2, letμ(G₁)=c^φ(E∩G₁) andν(G₂)=c^ψ(F∩G₂).

Thenμandν are finite Borel measures. IfG1⊂EandG2⊂F, thenμ(G1)=c^φ(G1), ν(G2)=c^ψ(G2) and (μ×ν)(G1×G2)=c^φ(G1)c^ψ(G2).

The inequalities lim sup

ε!0

(μ×ν)(B((x, y), ε))

(φ×ψ)(B((x, y), ε))≤lim sup

ε!0

μ(B(x, ε)) φ(B(x, ε))lim sup

ε!0

ν(B(y, ε)) ψ(B(y, ε)) are true for any (x, y)∈E×F.

From Corollary 3.3 there existE⊂EandF⊂Fsuch thatμ(E)=μ(E),ν(F)=

ν(F) and for all (x, y)∈E×F,d^φ_c(x, E)=d^ψ_c(y, F)=1. Therefore for all such (x, y), lim sup

ε!0

(μ×ν)(B((x, y), ε))

(φ×ψ)(B((x, y), ε))≤d_c^φ(x, E)d^ψ_c(y, F)≤1.

From (6) we deduce that (μ×ν)(E×F)≤c^φ×ψ(E×F). Since (μ×ν)(E×F)=

μ(E)ν(F)=c^φ(E)c^ψ(F) and c^φ×ψ(E×F)≤c^φ×ψ(E×F) we obtain the inequal- ity (15).

4. Related dimensions

LetEbe a bounded Borel set inR^D. Let us call ascale of functions associated with E a family Φ of determining functions φ: B_E!(0,∞), such that for any φ1

andφ₂ in Φ, only the three following cases can occur:

(a)φ1=φ2;

(b) for anyx∈E, lim_ε!0(φ1/φ2)(B(x, ε))=0;

(c) for anyx∈E, lim_ε!0(φ1/φ2)(B(x, ε))=+∞.

The measures c^φ(E) and p^φ(E) are significant only ifφbelongs to such a scale.

Then we may hope to associate a notion ofdimensionwith Φ. We choose to consider families Φ={φ_α:α∈R}parameterized by a real numberα. To be more specific, let us consider throughout this paper the following type of determining functions:

Assumption1. φ_α(B)=p(B)q(B)^α, wherep:B_E!(0,∞),q:B_E!(0,∞), and if

f(ε) = sup{q(B) :B∈ BE and diam(B)≤ε}, then lim_ε!0f(ε)=0.

This property ensures that Φ={pq^α:α∈R} is a scale of functions.

Notation. We will write c^αinstead of c^φαfor simplicity. Similarly, C^α, P^αand p^αreplace C^φα, P^φα and p^φα, respectively.

The classical theory deals with p=1 andq=diam, but we will consider other scales Φ in the sequel.

Proposition 4.1. Let α<β.

If C^α(E)<∞, thenC^β(E)=0;ifC^β(E)>0, thenC^α(E)=∞. The same results hold for c^α,P^α andp^α.

Proof. For every ε>0 and B∈B_E such that diam(B)≤ε, we have φ_α(B)≥ f(ε)^α−βφ_β(B). Therefore C^α_ε(E)≥f(ε)^α−βC^β_ε(E). Then letε!0.

Dimensions. We may therefore define the following dimensions:

dimΦ(E) = inf{α: c^α(E) = 0}; Δ_Φ(E) = inf{α: P^α(E) = 0}; Dim_Φ(E) = inf{α: p^α(E) = 0}.

The properties of dimΦ, ΔΦand DimΦderive easily from those of the related measures or pre-measures. In particular we have the following facts:

Since c^α, P^αand p^αare increasing set functions, dimΦ, ΔΦand DimΦare also increasing set functions.

Since P^αis subadditive, Δ_Φ isstable, that is

ΔΦ(E1∪E2) = max{ΔΦ(E1),ΔΦ(E2)}. (16)

Since c^αand p^αareσ-subadditive, dim_Φand Dim_Φareσ-stable, in other words dimΦ

n≥0

E_n = sup{dimΦ(E_n)}, DimΦ n≥0

E_n = sup{DimΦ(E_n)}. (17)

This allows us to define dimΦand Dim_φ on unbounded sets as well.

Since c^α(E)≤p^α(E)≤P^α(E),

dimΦ(E)≤DimΦ(E)≤ΔΦ(E).

(18)

From Δ_Φa direct definition of Dim_Φmay be derived:

Proposition 4.2. For any Borel set E, let ΔˆΦ(E) = inf

sup ΔΦ(E_n) :E_n are bounded Borel sets, withE=

n≥0

E_n

.

Then

ΔˆΦ(E) = DimΦ(E).

(19)

Proof. LetE_n be a sequence of bounded sets coveringE. Then DimΦ(E)≤sup

n≥0

DimΦ(E_n)≤sup

n≥0

ΔΦ(E_n).

This gives Dim_Φ(E)≤Δˆ_Φ(E).

For the other direction let α>DimΦ(E). Then p^α(E)=0, and there exists {E_n}n≥0such thatE=

n≥0E_nand

n≥0P^α(E_n)≤1. For everyn, P^α(E_n)≤1, so that ΔΦ(E_n)≤α. Therefore ˆΔΦ(E)≤α.

As for the measures c^α and p^α, a Borel measure μ spread onE may help to evaluate the value of the dimensions:

Theorem 4.3. Let μbe a finite orσ-finite Borel measure such that μ(E)>0, and

α_μ(B) =logμ(B)−logp(B) logq(B) . Then

x∈Einf lim inf

ε!0 α_μ(B(x, ε))≤dimΦ(E)≤sup

x∈Elim inf

ε!0 α_μ(B(x, ε)), (20)

x∈Einf lim sup

ε!0

α_μ(B(x, ε))≤DimΦ(E)≤sup

x∈Elim sup

ε!0

α_μ(B(x, ε)).

(21)

Proof. Without loss of generality, assume thatμ(E)<∞.

For (20), we make use of (6). Letα₁ be the left-hand side of (20), and assume that α1>0. Letβ∈(0, α1). For anyx∈E,

lim inf

ε!0 α_μ(B(x, ε))> β =⇒ lim sup

ε!0

μ(B(x, ε)) φ_β(B(x, ε))≤1.

The right inequality of (6) gives c^β(E)≥μ(E). Therefore dimΦ(E)≥β, so that dimΦ(E)≥α1.

Similar arguments are used to obtain the right inequality of (20) from the left inequality of (6). In the same way, (21) derives from (10).

For further application, let us deal with the sum of two determining functions:

Proposition 4.4. Let p₁ and p₂ be two functions :B_E!(0,∞), and q be as above. We consider the function scales

Φ1={p1q^α:α∈R}, Φ2={p2q^α:α∈R} and Φ1={(p1+p2)q^α:α∈R}.

Then

max{dim_Φ1(E),dim_Φ2(E)} ≤dim_Φ(E)≤Dim_Φ(E) = max{Dim_Φ1(E),Dim_Φ2(E)}. (22)

Proof. It suffices to show that

DimΦ(E)≤max{DimΦ₁(E),DimΦ₂(E)}.

For any determining functionsφandψ, the inequality P^φ+ψ≤P^φ+P^ψ is obtained through elementary arguments. This implies that

ΔΦ(E)≤max{ΔΦ₁(E),ΔΦ₂(E)}.

Let α>Dim_Φi(E) for i=1,2. We can find a cover {E_i}i≥0 of E such that sup ΔΦ₁(E_i)<α, and a cover {F_j}j≥0 of E such that sup ΔΦ₂(F_j)<α. Let G_ij= E_i∩F_j. Then {G_ij}i,j≥0 is a cover of E, and for all i, j, ΔΦ₁(G_ij)<α and ΔΦ₂(G_ij)<α. Therefore sup_i,jΔΦ(G_ij)≤α. This proves that DimΦ(E)≤α, and gives the desired inequality.

5. A special study of Δ_Φ

The set function ΔΦ becomes the well-known box dimension when φ_α= diam^α [12]. Among important features of this dimension are

(i) its definition using balls of equal diameter;

(ii) the equality Δ(E)=Δ(E), whereE is the closure ofE.

Letφ_α(B)=p(B)q(B)^α as before. With the help of some mild assumptions on pandqwe will be able to recover these properties.

The function f(ε) has been introduced in the last section. The following strengthens Assumption 1:

Assumption2. There exists two constants c0, α0>0 such that for all B∈BE, f(ε)≤c0ε^α0.

But we also need a few more conditions onpandq:

Assumption3. Both functions pandq areweakly increasing, in the following sense: There exists real constantsa1anda2such that for all balls BandB ofBE, B⊂B, diam(B)≤2 diam(B),

p(B)≤a1p(B) and q(B)≤a2q(B).

Withφ_α=pq^α, this inequality implies that φ_α(B)≤a1a^α₂φ_α(B).

(23)

From now on we will assume that Assumptions 2 and 3 are satisfied.

5.1. Packing with equal balls

In this sectionEis bounded. The following set function is defined like P^α_ε, with the only difference that the packings are made up with balls of equal diameter. Let

Q^α_ε(E) = sup

i≥0

φ_α(B_i) :B_i∈ BE are disjoint and diam(B_i) =ε

.

As a function ofε, Q^αdoes not need to be monotonous. Hence there is no limit as ε!0 in general. Let

Q^α(E) = lim sup

ε!0

Q^α_ε(E).

The statement of Proposition 4.1 is still true for Q: Ifα<β, then

Q^α(E)<∞ =⇒ Q^β(E) = 0 and Q^β(E)>0 =⇒ Q^α(E) =∞.

From that, we deduce the existence of a critical index, or dimension. We are willing to show that this dimension is precisely ΔΦ.

The inequality Q^α≤P^α is easily verified. But these two set functions are not equivalent in general.

Proposition 5.1. Let α<β. If P^β(E)>0, thenQ^α(E)=∞. We need the following elementary lemma:

Lemma 5.2. Let {B(x_i, ε)}_i≥0 be a family of disjoint balls, anda≥1. There exists an integerK1≤(2a+1)^Dsuch that the family{B(x_i, aε)}i≥0may be separated intoK1 families, each of them made up with disjoint balls.

Proof. Fori=j,B(x_i, aε)∩B(x_j, aε)=∅⇔B(x_j, ε)⊂B(x_i,(2a+1)ε). The ball B(x_i,(2a+1)ε) cannot contain more than (2a+1)^Ddisjoint balls of radiusε. Thus B(x_i, aε) cannot meet more than (2a+1)^D ballsB(x_j, aε). A standard argument achieves the proof.

Proof of Proposition 5.1. It suffices to show that Q^α(E)>0. Let η=β−α, ε∈(0,1), anda∈(0,P^β(E)). There exists anε-packingR={B_i}_i≥0 ofE such that

i≥0φ_β(B_i)≥a. For any non-negative integerk, let

Rk={B∈ R: 2^−k−1<diam(B)≤2^−k}.

When ε≤2^−k−1, we have Rk=∅. The Rk are such that

k≥0Rk=R, so that

k≥0

{φ_β(B):B∈Rk}≥a. Let c0 and α0 be as in Assumption 2. From the equality

a=

k≥0

a(1−2^−α0η)2^−kα0η we deduce the existence of an integerk0such that

{φ_β(B) :B∈ Rk0} ≥a(1−2^−α0η)2^−k0α0η. (24)

LetC_j=B(x_j, ε_j) be the balls ofRk0. LetC_j^∗=B(x_j,2^−k0−1) andC_j^∗∗=B(x_j,2ε_j).

Since 2^−k0−1<2ε_j≤2^−k0,C_j⊂C_j^∗⊂C_j^∗∗. As theC_j are disjoint, Lemma 5.2 implies that for some integerK1≤5^D, the family{C_j^∗∗}_j≥0may be divided intoK1families of disjoint balls. The same is true for the family{C_j^∗}j≥0, so that

j≥0

φ_α(C_j^∗)≤K1Q^α_2−k0(E).

(25)

For everyj,φ_β(C_j^∗)≤f(2^−k0)^ηφ_α(C_j^∗). Using Assumption 2, φ_β(C_j^∗)≤c^η₀2^−k0α0ηφ_α(C_j^∗).

Sincepandqare weakly increasing, (23) gives

φ_β(C_j)≤c₁2^−k0α0ηφ_α(C_j^∗) (26)

withc1=a1a^β₂c^η₀. Putting together (24) and (26) shows that a(1−2^−α0η)≤2^k0α0η

j≥0

φ_β(C_j)≤c₁

j≥0

φ_α(C_j^∗).

With (25) we obtain

Q^α₂−k0(E)≥a(1−2^−α0η) c1K1

.

The right-hand side is strictly positive and does not depend onε. Since 2^−k0<4ε, we deduce that lim sup_ε!0Q^α_ε(E)>0.

Corollary 5.3. ΔΦ(E)=inf{α:Q^α(E)=0}.

Proof. Ifα>ΔΦ(E), then P^α(E)=0, so that Q^α(E)=0.

If Q^α(E)=0, then for all β >α, P^β(E)=0, so that β≥Δ_Φ(E). Thereforeα≥ ΔΦ(E).