Curvature measures and fractals

Dissertation

zur Erlangung des akademischen Grades

doctor rerum naturalium (Dr. rer. nat.)

vorgelegt dem Rat der

Fakultat fur Mathematik und Informatik

der Friedrich-Schiller-Universitat Jena

von Dipl.-Math. Steffen Winter

geboren am 14. November 1977 in Potsdam

Gutachter

1. Prof. Dr. Martina Zahle

2. Prof. Dr. Michel Lapidus

Tag der letzten Prufung des Rigorosums: 31. Marz 2006

Tag der offentlichen Verteidigung: 7. April 2006

Zusammenfassung

In der vorliegenden Arbeit werden die Begriffe fraktale Krummungen und fraktale Krum-mungsmaße eingefuhrt mit der Intention, charakteristische Großen bzw. geometrischeMaße fur fraktale Mengen bereitzustellen, die die klassischen Krummungsbegriffe, diefur solche Mengen oft nicht erklart oder nicht sinnvoll sind, ersetzen konnen.

Die neuen Großen werden aus den klassischen im wesentlichen folgendermaßen abgelei-tet: Krummungen und Krummungsmaße sind zwar gewohnlich fur eine fraktale MengeF ⊂ R

d selbst nicht erklart, jedoch oft fur die ǫ-Parallelmengen von F , d.h. fur dieMengen

Fǫ =

x ∈ Rd : d(x, F ) ≤ ǫ

.

Laßt man nun ǫ gegen Null konvergieren und reskaliert geeignet, so erhalt man diefraktalen Krummungen als Grenzwert der klassischen. Ist etwa Ck(Fǫ) die k-te totaleKrummung von Fǫ, so ist die k-te fraktale Krummung von F gegeben durch

Cfk (F ) := lim

ǫ→0ǫskCk(Fǫ) ,

falls der Grenzwert existiert, wobei der Skalierungsexponent sk = sk(F ) zunachst ge-eignet zu wahlen ist. Ahnlich werden fraktale Krummungsmaße als in derselben Weisereskalierte schwache Grenzwerte der klassischen Krummungsmaße der Parallelmengeneingefuhrt. Der bekannte Minkowski-Inhalt und die kurzlich eingefuhrten fraktalenEulerzahlen ordnen sich als Spezialfalle in dieses Konzept ein.

Als eine erste Anwendung des Konzeptes werden hier die fraktalen Krummungenund Krummungsmaße selbstahnlicher Mengen untersucht. Unter Voraussetzung deroffenen-Mengen-Bedingung wird, ausgehend von den klassischen Krummungsmaßen imKonvexring, sowohl die Existenz fraktaler Krummungen als auch die Konvergenz derKrummungsmaße gezeigt. Dabei ist fur manche Mengen (d.h. im arithmetischen Fall)eine zusatzliche Mittelung der Grenzwerte erforderlich. Mit dieser Untersuchung hof-fen wir, einerseits einen Beitrag zur Diskussion um eine weitere Verallgemeinerung derKrummungsbegriffe zu leisten, andererseits mit den fraktalen Krummungen Großen be-reitzustellen, die der praktischen Klassifizierung, Beschreibung und Unterscheidung frak-taler Strukturen dienlich sind.

3

Contents

Introduction 7

1. Main results and examples 11

1.1. The concept of fractal curvatures . . . . . . . . . . . . . . . . . . . . . . 111.2. Fractal curvatures for self-similar sets . . . . . . . . . . . . . . . . . . . . 141.3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.4. Fractal curvature measures . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2. Curvature measures in the convex ring 35

2.1. Properties of curvature measures . . . . . . . . . . . . . . . . . . . . . . . 352.2. Variation measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3. Adapted Renewal theorem 43

4. Fractal curvatures – proofs 47

4.1. Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2. Proof of Theorem 1.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3. Convex representations of Fǫ . . . . . . . . . . . . . . . . . . . . . . . . . 524.4. Cardinalities of level set families . . . . . . . . . . . . . . . . . . . . . . . 554.5. Bounds for the total variations . . . . . . . . . . . . . . . . . . . . . . . . 584.6. Proof of Gatzouras’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 60

5. Fractal curvature measures – proofs 63

5.1. A separating class for F . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2. Proof of Theorem 1.4.1 and 1.4.2 . . . . . . . . . . . . . . . . . . . . . . . 655.3. Proof of Theorem 1.4.3 and 1.4.4 . . . . . . . . . . . . . . . . . . . . . . . 70

A. Appendix: Signed measures and weak convergence 73

Bibliography 75

5

Introduction

There were two main incentives for working on the subject presented in this thesis.On the one hand there is the long standing quest in classical geometry to extend thenotions of curvature as far as possible. Curvatures and Curvature measures have beenidentified for many classes of sets as in a sense canonical objects for the description oftheir geometry. On the other hand there is the desire to understand the geometry offractal sets better, in particular, to find quantitative measures to describe the geometricfeatures of fractals. In this thesis it is intended to make a contribution to both of thesegoals by introducing the notion of fractal curvatures.

A milestone in the development of curvature measures was their introduction for setswith positive reach by Herbert Federer in his famous paper [8], which joined and uni-fied the two existing notions of curvature for differentiable manifolds and convex bodies.Federer’s work effected the further development of the formerly independent parts ofthe theory and led for instance to an axiomatic characterization of curvature measuresof convex sets by Rolf Schneider [28] and their additive extension to the convex ring byHelmut Groemer [12]. It was also the starting point for further generalisations and ex-tensions of the notion of curvature measures for instance to unions of sets with positivereach and Lipschitz manifolds by Martina Zahle [32, 33, 34], Jan Rataj and MartinaZahle [24, 25, 26] and Joseph H. G. Fu [10]. In a very recent paper by Daniel Hug,Gunther Last und Wolfgang Weil [14], curvature measures were introduced for generalclosed sets, which marks the tentative culmination of this generalisation process. In con-trast to the previously mentioned curvature measures they are non-additive. However,since fractal sets are usually considered as being closed, one could feel tempted to seethose curvature measures as the endpoint and solution to the above raised questions.Unfortunately, it seems that, although these measures are defined for fractal sets, theyare not capable of sufficiently enlightening the ”fractal” features of their structure.

In fractal geometry on the other hand, apart from the different concepts of dimensionthere have been up to now very few generally accepted quantitative measures, which areable to provide additional information on the geometric structure of fractal sets. One ofthem is the Minkowski content which was proposed by Benoit Mandelbrot in [23] as ameasure of lacunarity, i.e. as a characteristic for the porosity of fractal sets. In the samepaper he also remarked the insufficiency of having just one such measure available:

”But, in general, one must not expect the search for a measure of lacunarityto converge to a single number, because the task of specifying a generaltexture numerically is at least as difficult as the task of specifying the shapeof a curve.”

The Minkowski content is derived from the classical notion of volume or Lebesgue mea-sure by a limiting procedure. Instead of looking at the volume of a compact set F ⊂ R

d

7

itself – which, in case F is a fractal, typically is zero and thus not interesting – onestudies the volume of its ǫ-parallel sets

Fǫ =

x ∈ Rd : d(x, F ) ≤ ǫ

,

and, in particular, how the volume evolves as ǫ tends to zero. If the volume λd(Fǫ)is rescaled appropriately, namely with the factor ǫm−d, where m = m(F ) denotes theMinkowski dimension (or box dimension) of F , often some limit exists and is then termedthe Minkowski content of the set F :

M(F ) := limǫ→0

ǫm−dλd(Fǫ).

We introduce fractal curvatures in the same way as the Minkowski content replacingthe volume by some curvature measure. Supposed curvature measures C0(Fǫ, · ), . . . ,Cd(Fǫ, · ) are defined for the parallel sets Fǫ of F , we consider first for each k = 0, . . . , dthe total curvatures Ck(Fǫ) := Ck(Fǫ, R

d). The k-th fractal curvature of F is the limit

Cfk (F ) := lim

ǫ→0ǫskCk(Fǫ),

provided this limit exists. Here the right choice of the scaling exponent sk is one of thecentral questions. Due to the nature of curvature measures, the Minkowski content ap-pears as a special case in this concept. It coincides with the d-th fractal curvature, sinceCd(Fǫ, · ) = λd(Fǫ ∩ · ). In [22], fractal Euler numbers were studied by Marta Llorenteand the author, which are derived in a similar manner from the Euler characteristic ofthe parallel sets. These numbers fit into the presented framework too. They are closelyrelated to the 0-th fractal curvature.

The concept of fractal curvatures comes up very naturally not only because of thedirect analogy with the Minkowski content. It is based on parallel sets, which playan important role for the definition of ”classical” curvature measures via Steiner typeformulas too. Moreover, under certain assumptions on F , for instance if F is convex, the”classical” curvatures Ck(Fǫ) (and curvature measures Ck(Fǫ, · )) are known to convergeto the corresponding curvature Ck(F ) (or curvature measure Ck(F, · ), respectively) ofF as ǫ approaches zero. The new feature in the fractal case is the need to rescale thecurvatures Ck(Fǫ).

In the present exposition we restrict ourselves to curvature measures as defined in theconvex ring, i.e. we assume throughout that the parallel sets Fǫ (but not the fractal setsF themselves, of course) are finite unions of convex sets. We are aware of the loss ofgenerality this restriction implies, but this will keep the presentation simpler and allowconcentration on the new ideas. Nevertheless, we are convinced that fractal curvaturescan be introduced in a more general setting. This investigation is just a starting point.

We apply the concept to self-similar sets. We will show that for most self-similarsets satisfying the open set condition, fractal curvatures exist, provided that they arenot too regular (i.e. provided they are non-arithmetic). Since many popular examplesof self-similar sets like the Sierpinski gasket are in fact very regular, it turns out to beuseful to study averaged limits of total curvatures as well. Average fractal curvaturesare shown to exist also for regular self-similar sets. The results on the existence of

8

total curvatures are quite analogical to known results for the Minkowski content (cf. inparticular Dimitris Gatzouras [11]) and are derived in a similar way by application ofthe Renewal Theorem to some appropriate renewal equation. We want to propose thesequantities as geometric characteristics for fractal sets, which could play a similar role forfractal sets as the classical total curvatures (or equivalently volume, surface area, Eulercharacteristic and so on) in classical geometry. Therefore we will also demonstrate inseveral examples how fractal curvatures are determined and interpreted.

Up to this point the analogies with the Minkowski content were a good guide to leadus. But we will go one step further and study not only the global but also the locallimiting behaviour. Fractal curvatures are limits of the total curvatures of the parallelsets like the Minkowski content is a limit involving the ”total” volume of the parallelsets. This does not take into account the fact that curvature measures are geometricmeasures describing the local behaviour of the sets. Naturally the question appears,what happens to the curvature measures Ck(Fǫ, · ) as ǫ tends to zero. For self-similarsets F , it turns out that, if the curvature measures are rescaled appropriately, limitscan be obtained in the sense of weak convergence. Again some averaging does improvethe convergence behaviour for very regular sets. The derived limit measures could beregarded as fractal curvature measures of the initial set F . They turn out to be self-similar measures. Each fractal curvature measure is some multiple of the Hausdorffmeasure on F and its total mass is given by the corresponding fractal curvature. Thegeometric interpretation of this convergence is as follows: If B is a ”nice” set, then therescaled curvature ǫskCk(Fǫ, B) of Fǫ in the set B converges to the corresponding ”fractalcurvature” of F in B.

In the case k = d, this weak convergence result extends the known results for theMinkowski content. It provides a local characterization of the limiting behaviour of thevolume of the parallel sets. In this statement the assumption on the parallel sets ofbeing finite unions of convex sets can be dropped. It is shown to be valid for arbitraryself-similar sets satisfying the open set condition.

Summing up, we introduce fractal curvatures, which we propose as quantitative mea-sures for the geometry of fractal sets, and fractal curvature measures, which we regardas a replacement of classical curvature measures in the fractal world. Results on theirexistence and their explicit characterization are obtained for certain classes of self-similarsets in R

d.The thesis is organized as follows. In Chapter 1 the concept of fractal curvatures is

introduced and all the main results are presented. Most of the proofs are postponedto later Chapters. Section 1.1 provides the setting and the main definitions, while in1.2 self-similar sets are introduced and fractal curvatures are studied for those sets. InSection 1.3, we discuss several examples to illustrate the results of the preceding sectionand show how fractal curvatures are practically computed and interpreted. Then we turnto the localisation and in Section 1.4 study weak limits of rescaled curvature measures.

Chapter 2 provides a brief introduction to curvature measures in the convex ring.We summarize their most important properties and collect in Section 2.2 a number ofstatements about their variation measures. In Chapter 3 we make the first steps towardsthe proofs of the main results by discussing the Renewal Theorem and reformulating itin a way which is most convenient for us. Chapter 4 is devoted to the proofs of theresults of Section 1.2 on fractal curvatures and the associated scaling exponents, while

9

in Chapter 5 the results of Section 1.4 are proved. An Appendix recalls some facts aboutsigned measures and especially the notion of weak convergence of signed measures, whichhas seldom been used in the literature.

I am most grateful to all those who helped me along the way of writing this thesis withtheir encouragement and support, with stimulating discussions or valuable commentsand suggestions. Especially, I wish to thank Professor Martina Zahle for her excellentsupervision of my research during three years and for the confidence she has shown inme and my work.

10

1. Main results and examples

1.1. The concept of fractal curvatures

Convex ring. A set K ⊆ Rd is said to be convex if for any two points x, y ∈ K the line

segment connecting them is a subset of K. Denote by Kd the family of all convex bodies,i.e. of all nonempty compact convex sets in R

d. A set K is called polyconvex if it can berepresented as a finite union of convex bodies. The class Rd of all polyconvex sets in R

d

is called the convex ring. It is stable with respect to finite unions and intersections.

Curvature measures. For each set K ∈ Rd there exist d + 1 totally finite signedmeasures C0(K, · ), C1(K, · ), . . . , Cd(K, · ), called the curvature measures of K, whichdescribe the local geometry of the set K. For convex bodies K, they are characterizedby the Local Steiner formula and by additive extension they are generalized to arbitrarypolyconvex sets K. The total mass Ck(K) := Ck(K, Rd) of the measure Ck(K, · ) iscalled the k-th total curvature of K. It is also known as k-th intrinsic volume of K.Curvature measures of polyconvex sets are well understood and have many advantageousproperties, including covariance with respect to Euclidean motions, homogeneity, localityand additivity. In Chapter 2 we give a more detailed introduction to them and alsohints to the literature. For the moment we just add a few words on their geometricmeaning. Cd(K, · ) is nothing but the volume restricted to K, i.e. the d-dimensionalLebesgue measure λd(K ∩ · ), while Cd−1(K, · ) is half the surface area of K, providedthat K is the closure of its interior, K = intK. Both measures are always positive incontrast to the situation for k = 0, . . . , d − 2, for which Ck(K, · ) is a signed measurein general. Except for k = d, Ck(K, · ) is concentrated on the boundary ∂K of K. Forconvex bodies, Ck(K, · ) has an interpretation in terms of the k-dimensional volumes ofthe projections of K to k-dimensional linear subspaces. More precisely, Ck(K) is theaverage of these volumes over all subspaces. In general, the numbers Ck(K, B) describethe different aspects of the ”curvedness” of ∂K inside the set B. Finally, by the GaußBonnet formula, the 0-th total curvature C0(K) coincides with the Euler characteristicof K, the topological invariant defined in algebraic topology.

The positive, negative and total variation measure C+k (K, · ), C−

k (K, · ) and Cvark (K, · )

of the signed measure Ck(K, · ) are defined respectively, by setting for each Borel setA ⊆ R

d

C+k (K, A) := sup

B⊆ACk(K, B) and C−

k (K, A) := − infB⊆A

Ck(K, B)

andCvar

k (K, A) := C+k (K, A) + C−

k (K, A).

The variations are positive measures and satisfy the relation

Ck(K, · ) = C+k (K, · ) − C−

k (K, · ), (1.1.1)

11

called the Jordan decomposition of Ck(K, · ). Positive and negative variation measuresare useful for localizing ”positive” and ”negative” curvature or, more figuratively, to dis-tinguish locally convexity from concavity. Some of the properties of curvature measurescarry over to their variation measures. For more details we refer to the discussion inSection 2.2.

Central assumption. Fractal sets F , which we are interested in here, are usually notpolyconvex. Their curvature measures Ck(F, · ) are typically not defined in any classicalsense. To investigate their geometric properties, we study the curvature measures oftheir parallel sets. For a compact set F ⊂ R

d and ǫ > 0, the ǫ-parallel set of F is the set

Fǫ :=

x ∈ Rd : d(x, F ) ≤ ǫ

,

where d(x, F ) = infy∈F d(x, y) denotes the Euclidean distance between x and F . Inparticular, we want to study how the curvature measures of the parallel sets Fǫ behaveas ǫ tends to zero. In general, curvature measures need not be defined for the parallelsets. Therefore, unless indicated otherwise, throughout the paper we make the followingassumption:

Assume that Fǫ ∈ Rd for all ǫ > 0 .

The supposition of F having polyconvex parallel sets ensures the existence of the d + 1curvature measures C0(Fǫ, · ), C1(Fǫ, · ), . . . , Cd(Fǫ, · ) for each parallel set Fǫ. In orderto introduce the concepts below it is essential to have curvature measures defined for theparallel sets. These need not necessarily be the curvature measures from the convex ringsetting. Generalizations (in particular of the Definitions 1.1.2, 1.1.5 and 1.1.6 below) arepossible, for instance to sets whose parallel sets are unions of sets with positive reach.For simplicity, we restrict ourselves to the polyconvex setting here. One particular reasonis the property of polyconvex sets to have polyconvex parallel sets. If Fǫ ∈ Rd for someǫ > 0, then Fǫ+δ ∈ Rd for all δ > 0. Therefore, the existence of arbitrary small ǫ > 0such that Fǫ ∈ Rd is already sufficient to ensure that all parallel sets are polyconvex.Conversely, if there exist some ǫ0 > 0 such that Fǫ0 is not polyconvex, then the sameholds for all smaller parallel sets of F .

Fact 1.1.1. Either Fǫ ∈ Rd for all ǫ > 0 or there exists ǫ0 > 0 such that Fǫ /∈ Rd for all0 < ǫ ≤ ǫ0.

For self-similar sets we even have the dichotomy that either all or none of their parallelsets are polyconvex (cf. Proposition 1.2.1).

Scaling exponents. We first concentrate on the total curvatures Ck(Fǫ) = Ck(Fǫ, Rd)

of the parallel sets Fǫ. The typical behaviour of Ck(Fǫ) as ǫ → 0 is either to converge tozero or to tend to ±∞. To obtain more information about the limiting behaviour somerescaling is advisable. We study the expressions ǫtCk(Fǫ) where the exponent t ∈ R hasto be chosen appropriately for each k (and F ). If t ∈ R is chosen too small, ǫtCk(Fǫ) willtend to ±∞, while ǫtCk(Fǫ) tends to 0 if t is too large. So the exponent should be at theborderline between the two extremes. Taking the infimum over all t for which ǫtCk(Fǫ)approaches zero or the supremum over those t for which ǫt |Ck(Fǫ)| is unbounded seems

12

a reasonable choice for the exponent, especially if both numbers happen to coincide. Butwe have to take the local character of curvature measures into account and the fact thatthey are, in general, signed measures. The total mass Ck(Fǫ) can be zero, while at thesame time locally the measure Ck(Fǫ, · ) is very large. The positive curvature in somepart can equal out the negative curvature in some other part of the set to give totalmass zero. Therefore it is more reasonable to use the total variation for the definitionof the scaling exponent.

Definition 1.1.2. Let F ⊆ Rd a compact set with polyconvex parallel sets, and let

k ∈ 0, 1, . . . , d. The k-th curvature scaling exponent of F is the number

sk = sk(F ) := inft : ǫtCvar

k (Fǫ) → 0 as ǫ → 0

.

sk is well defined at least in the sense of being an element of R = R ∪ −∞, +∞.If lim infǫ→0 ǫskCvar

k (Fǫ) > 0, then, clearly, sk is the only interesting exponent forthis expression, since, for all t > sk, ǫtCvar

k (Fǫ) → 0 as ǫ → 0 and for all t < sk,ǫtCvar

k (Fǫ) → +∞. Then sk is equivalently characterized by the expression sk :=sup

t : ǫtCvar

k (Fǫ) → ∞ as ǫ → 0. In general, sk need not coincide with sk (but al-

ways sk ≤ sk) and one should then better speak of lower and upper scaling exponents,respectively. However, here we will only bother about sk. Observe that sk(F ) is invariantwith respect to Euclidean motions and scaling.

Remark 1.1.3. (i) Since Cvard (Fǫ) = Cd(Fǫ) = λd(Fǫ), the number d−sd(F ) corresponds

to the upper Minkowski dimension (or box dimension) of F , which is defined moregenerally, namely for arbitrary (compact) sets.

(ii) In [22], Marta Llorente and the author introduced the Euler exponent σ = σ(F )of F as the infimum inf

t ≥ 0 : ǫt |χ(Fǫ)| → 0 as ǫ → 0

, where χ denotes the Euler

characteristic. σ(F ) is defined for more general sets F than those discussed here. Forsets F with Fǫ ∈ Rd for all ǫ > 0, however, the Euler exponent is closely related tos0(F ), since, by the Gauss Bonnet formula, C0(Fǫ) = χ(Fǫ). The main difference isthat in the definition of σ(F ) we work with absolute values |C0(Fǫ)| = |χ(Fǫ)|, whilefor s0(F ) we used the total variations Cvar

0 (Fǫ). Often both exponents coincide, butsometimes they differ. This corresponds very well to the different geometric meaning ofχ(Fǫ) as a topological invariant and C0(Fǫ, · ) as a curvature measure. For the differentinterpretations of σ and s0 confer Example 1.3.5. Note that in general σ(F ) ≤ s0(F ) .

Remark 1.1.4. By replacing Cvark (Fǫ) in the definition of sk with C+

k (Fǫ) and C−k (Fǫ),

scaling exponents s+k and s−k can be defined. Since Cvar

k (Fǫ) = C+k (Fǫ) + C−

k (Fǫ), theysatisfy the relation sk = max

s+k , s−k

. Hence one of these two exponents must always

coincide with sk, while the second one can be smaller. This does in fact happen as wewill see later. (cf. Remark 1.3.6)

Fractal curvatures. Having defined the scaling exponents for total curvatures, we cannow ask for the existence of rescaled limits.

Definition 1.1.5. Let k ∈ 0, 1, . . . , d. If the limit

Cfk (F ) := lim

ǫ→0ǫskCk(Fǫ)

exists, then it is called the k-th fractal (total) curvature of the set F .

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Unfortunately, the existence of such limits is not always ensured. From the study oflocal quantities of self-conformal sets like densities or tangent measure distributions itis well-known that, in general, average limits provide better results. It turns out thatthis is a useful tool for our purposes too.

Definition 1.1.6. Let k ∈ 0, 1, . . . , d. If the limit

Cfk(F ) := lim

δ→0

1

|log δ|

∫ 1

δǫskCk(Fǫ)

dǫ

ǫ

exists, then it is called the k-th average fractal (total) curvature of the set F .

In both definitions one could also work with upper and lower limits. However, herewe concentrate on the existence of the (average) limits. Note that whenever the limit inDefinition 1.1.5 exists then the corresponding average limit exists as well and coincideswith the limit. Moreover, (average) fractal curvatures have the following propertieswhich are due to the corresponding properties of the total curvatures:

Motion invariance: If g is an Euclidean motion, then Cfk (gF ) = Cf

k (F ).

Scaling property: If λ > 0 and λF := λx : x ∈ F, then Cfk (λF ) = λsk+kCf

k (F ).

Consistency: If F ∈ Rd, then Cfk (F ) exists and coincides with Ck(F ).

The consistency with the classical total curvatures is easily derived for convex sets fromthe continuity of curvature measures and in general from the additivity. All propertieshold likewise for the average fractal curvatures.

1.2. Fractal curvatures for self-similar sets

We want to apply the above concepts to study self-similar sets. In this section weobtain some results concerning the existence of (average) fractal curvatures of thesesets. The main statements are presented without proofs here. The proofs will be givenin Chapter 4.

Self-similar sets. Let Si : Rd → R

d, i = 1, . . . , N , be contracting similarities. Denotethe contraction ratio of Si by ri ∈ (0, 1) and set rmax := maxi=1,...,N ri and rmin :=mini=1,...,N ri. It is a well known fact in fractal geometry (cf. Hutchinson [15]), that forsuch a system S1, . . . , SN of similarities there is a unique, non-empty, compact subsetF of R

d such that S(F ) = F , where S is the set mapping defined by

S(A) =N⋃

i=1

SiA, A ⊆ Rd.

F is called the self-similar set generated by the system S1, . . . , SN. Moreover, theunique solution s of

∑Ni=1 rs

i = 1 is called the similarity dimension of F . The systemS1, . . . , SN is said to satisfy the open set condition (OSC) if there exists an open,non-empty, bounded subset O ⊂ R

d such that⋃

i SiO ⊆ O and SiO ∩ SjO = ∅ for alli 6= j. In [1], O was called a feasible open set of the Si, or of F , which we adopt here. For

14

Figure 1.1.: The Sierpinski gasket has polyconvex parallel sets, while the Cantor set onthe right does not.

convenience, we also say that F satisfies OSC, always having in mind a particular systemof similarities generating F . If additionally O ∩ F 6= ∅ holds for some feasible open setO of F , then S1, . . . , SN (or F ) is said to satisfy the strong open set condition (SOSC).It was shown by Andreas Schief in [27] that in R

d SOSC is equivalent to OSC, i.e. if Fsatisfies OSC, then there exists always some feasible open set O such that O ∩ F 6= ∅.It is easily seen that F ⊆ O for each feasible open sets O of F .

Let h > 0. A finite set of positive real numbers y1, ..., yN is called h-arithmetic ifh is the largest number such that yi ∈ hZ for i = 1, . . . , N . If no such number h existsfor y1, ..., yN, the set is called non-arithmetic. We attribute these properties to thesystem S1, . . . , SN or to F if and only if the set − log r1, . . . ,− log rN has them. Inthis sense, each set F (generated by some S1, . . . , SN) is either h-arithmetic for someh > 0 or non-arithmetic.

Parallel sets of self-similar sets. In order that the notions we introduced above aredefined for a self-similar set, we have to make sure that its parallel sets are polyconvex.It turns out that not all self-similar sets F have this property. But at least there is thedichotomy that either all or none of the parallel sets of F are polyconvex. This wasalready shown in [22].

Proposition 1.2.1. Let F a self-similar set. If Fǫ ∈ Rd for some ǫ > 0, then Fǫ ∈ Rd

for all ǫ > 0.

Therefore it suffices to check for an arbitrary parallel set Fǫ, whether or not it ispolyconvex, to know that for all parallel sets of F . For completeness, a simple proofof this statement is included in Section 4.1. Sierpinski gasket and Sierpinski carpet areself-similar sets with polyconvex parallel sets, while for instance the von Koch curve orthe Cantor set in Figure 1.1 do not have this property. In R all self-similar sets havepolyconvex parallel sets, since for each ǫ > 0, Fǫ ∈ R consists of a finite number ofintervals. In the sequel assume that Fǫ ∈ Rd for some (and thus all) ǫ > 0.

Scaling exponents of self-similar sets. The first question that arises is the one for theright scaling exponents sk(F ) of F . The following result provides an upper bound forthe k-th scaling exponent sk(F ).

15

Theorem 1.2.2. Let F be a self-similar set satisfying OSC and Fǫ ∈ Rd, and letk ∈ 0, . . . , d. The expression ǫs−kCvar

k (Fǫ) is uniformly bounded in (0, 1], i.e. there isa constant M such that for all ǫ ∈ (0, 1], ǫs−kCvar

k (Fǫ) ≤ M .

The proof is postponed to Section 4.5. The stated boundedness of the expressionǫs−kCvar

k (Fǫ) has the following immediate implications.

Corollary 1.2.3. sk ≤ s − k

Proof. By Theorem 1.2.2, lim supǫ→0 ǫs−kCvark (Fǫ) ≤ M and thus for each t > 0,

limǫ→0 ǫs−k+tCvark (Fǫ) ≤ limǫ→0 ǫt lim supǫ→0 ǫs−kCvar

k (Fǫ) = 0.

Corollary 1.2.4. The expression ǫs−k |Ck(Fǫ)| is bounded in (0, 1] by M .

Proof. Observe that |Ck(Fǫ)| ≤ Cvark (Fǫ) for each ǫ > 0.

Corollary 1.2.3 provides the upper bound s− k for the k-th scaling exponent sk(F ) andraises the question whether the equality sk = s− k holds. It will turn out that for mostself-similar sets (and most k) indeed sk = s − k. Unfortunately, this is not always thecase as the following example shows.

Example 1.2.5. The unit cube Q = [0, 1]d ⊂ Rd is a self-similar set generated for

instance by a system of 2d similarities each with contraction ratio 12 , which has similarity

dimension s = d. For the curvature measures of its parallel sets no rescaling is necessary.Since Q is convex, its parallel sets Qǫ are and so the continuity implies that, for k =0, . . . , d, Ck(Qǫ, · ) → Ck(Q, · ) as ǫ → 0. Therefore, sk(Q) = 0, which for k < d iscertainly different to d − k.

We investigate now the limiting behaviour of the expression ǫs−kCk(Fǫ) as ǫ → 0.Since such degenerate examples like the cubes exist, we can not expect that this willalways be the right expression to study.

Scaling functions. For a self-similar set F and k ∈ 0, . . . , d, define the k-th curvaturescaling function Rk : (0,∞) → R by

Rk(ǫ) = Ck(Fǫ) −N∑

i=1

1(0,ri](ǫ)Ck((SiF )ǫ). (1.2.1)

The function Rk allows to formulate a renewal equation so that the required informa-tion on the limiting behaviour of the expression ǫs−kCk(Fǫ) can be obtained from theRenewal theorem. Therefore it is essential to understand its properties. Geometricallythe meaning of Rk(ǫ) is the following: Since Fǫ =

⋃Ni=1(SiF )ǫ, the additivity of the total

curvatures implies that, for small ǫ (i.e. ǫ ≤ rmin), Rk describes the curvature of theintersections of the sets (SiF )ǫ:

Rk(ǫ) =∑

#I≥2

(−1)#I−1Ck

(⋂

i∈I

(SiF )ǫ

), (1.2.2)

where the sum is taken over all subsets I of 1, . . . , N with at least two elements. Hencethe function Rk describes the k-th curvature of what we call the overlap of Fǫ, namelythe set

⋃i6=j(SiF )ǫ ∩ (SjF )ǫ.

16

Existence of fractal curvatures. If the scaling function Rk is well behaved, which isthe case for sets satisfying OSC, then usually (average) k-th fractal total curvaturesexist. The precise statement is derived from the following result, which characterizesthe limiting behaviour of ǫs−kCk(Fǫ).

Theorem 1.2.6. Let F be a self-similar set satisfying OSC and Fǫ ∈ Rd. Then fork ∈ 0, . . . , d the following holds:

(i) The limit limδ→0

1|log δ|

∫ 1δ ǫs−kCk(Fǫ)

dǫǫ exists and equals to the finite number

Xk =1

η

∫ 1

0ǫs−k−1Rk(ǫ) dǫ, (1.2.3)

where η = −∑Ni=1 rs

i log ri.

(ii) If F is non-arithmetic, then the limit limǫ→0

ǫs−kCk(Fǫ) exists and equals Xk.

The proof of this statement will be given in Section 4.2. The number Xk defined in(1.2.3) as an integral over the function Rk determines the average limit – and in thenon-arithmetic case as well the limit – of the expression ǫs−kCk(Fǫ). If for F we had

that sk = s− k, then, by definition, the average fractal curvature Cfk(F ) would coincide

with Xk, and in case of a non-arithmetic set F also the fractal curvature Cfk (F ). But

this is not always true as we have seen in Example 1.2.5, where for some k, sk wasstrictly smaller than s − k. Additional assumptions are required. A sufficient conditionfor sk = s − k is the requirement that Xk is different from 0.

Corollary 1.2.7. If Xk 6= 0, then sk = s − k. Consequently, Cfk(F ) = Xk and, in case

F is non-arithmetic, also Cfk (F ) = Xk.

Proof. The assumption Xk 6= 0 and (i) of Theorem 1.2.6 imply that

0 < |Xk| =

∣∣∣∣limδ→0

1

|log δ|

∫ 1

δǫs−kCk(Fǫ)

dǫ

ǫ

∣∣∣∣ ≤ limδ→0

1

|log δ|

∫ 1

δǫs−k |Ck(Fǫ)|

dǫ

ǫ

≤ lim supǫ→0

ǫs−k |Ck(Fǫ)| ≤ lim supǫ→0

ǫs−kCvark (Fǫ)

and thus for all t > 0,

lim supǫ→0

ǫs−k−tCvark (Fǫ) ≥ |Xk| lim

ǫ→0ǫ−t = ∞.

Hence there is no t > 0 such that ǫs−k−tCvark (Fǫ) → 0 implying sk = s − k.

Formula (1.2.3) provides an explicit way to calculate Xk and therefore (average) fractalcurvatures, at least if Xk turns out to be different from 0. For k = d it can be shown thatalways Xd > 0 and thus sd = s − d. This case is related to the Minkowski content andwill be discussed separately below. So assume for the moment that k ∈ 0, . . . , d − 1.

For those k it remains to clarify the situation when Xk = 0 for a self-similar set. Firstnote that the condition Xk 6= 0 is not necessary for sk to be s − k. For Xk = 0 both

17

situations are possible - either sk < s−k or sk = s−k. In Example 1.3.5 we will discussa set of the latter type, while the cubes presented in Example 1.2.5 above are of theformer type. Note that in the latter case, Theorem 1.2.6 provides the right values for

the (average) k-th fractal curvature, namely Cfk(F ) = Xk = 0 and in the non-arithmetic

case as well Cfk (F ) = 0.

The next Theorem provides a tool to detect sets of the latter type, i.e. with sk = s−k(with or without Xk = 0). Here it is necessary to work locally rather then just with thetotal curvatures. For ǫ > 0, define the inner ǫ-parallel set of a set A by

A−ǫ := x ∈ A : d(x, Ac) > ǫ (1.2.4)

or equivalently as the complement of the (outer) ǫ-parallel set of the complement of A,i.e. A−ǫ = ((Ac)ǫ)

c. Inner parallel sets only make sense for sets with nonempty interior,otherwise they are empty.

Theorem 1.2.8. Let F be a self-similar set satisfying OSC and Fǫ ∈ Rd, O somefeasible open set of F , and k ∈ 0, . . . , d. Suppose there exist some constants ǫ0, β > 0and some Borel set B ⊂ O−ǫ0 such that

Cvark (Fǫ, B) ≥ β

for each ǫ ∈ (rminǫ0, ǫ0]. Then for all ǫ < ǫ0

ǫs−kCvark (Fǫ) ≥ c,

where c := βǫs−k0 rs

min > 0 .

The rough idea is that curvature in some advantageous location B in a large parallelset Fǫ, is exponentiated and spreaded by the self-similarity as ǫ tends to zero. A thoroughproof of this statement is provided in Section 4.5. An immediate consequence is that,under the hypotheses of Theorem 1.2.8, s − k is a lower bound for sk and thus

Corollary 1.2.9. sk = s − k

Proof. Theorem 1.2.8 implies that lim infǫ→0 ǫs−kCvark (Fǫ) ≥ c and thus for each t > 0,

lim infǫ→0 ǫs−k−tCvark (Fǫ) ≥ c limǫ→0 ǫ−t = ∞. Hence sk ≥ s − k. The validity of the

reversed inequality has already been obtained in Corollary 1.2.3.

Theorem 1.2.8 can be seen as a counterpart to Theorem 1.2.2. While the latterprovides an upper bound for the scaling exponent sk which is in a sense universal, theformer gives the corresponding lower bound, though only under additional assumptions.It is a useful supplement to Corollary 1.2.7 for the treatment of self-similar sets withXk = 0. Its power will be revealed in Example 1.3.5.

Minkowski content. For the case k = d the above results hold in a more general setting,namely without the assumption of polyconvex parallel sets. For this case the results arealready known. They have been obtained by Dimitris Gatzouras in [11]. We want torecall Gatzouras’s results and discuss more carefully how they fit into our setting.

18

First recall that Cd(Fǫ, · ) = λd(Fǫ ∩ · ), whenever Cd(Fǫ, · ) is defined. Therefore itis straightforward to generalize the definitions of the d-th scaling exponent and the d-th(average) fractal curvature to arbitrary compact sets F ∈ R

d. The resulting quantitiesare

sd = sd(F ) := inft : ǫtλd(Fǫ) → 0 as ǫ → 0

,

M(F ) := limǫ→0

ǫsdλd(Fǫ)

and

M(F ) := limδ→0

1

|log δ|

∫ 1

δǫsdλd(Fǫ)

dǫ

ǫ.

Obviously, d − sd coincides with the (upper) Minkowski dimension, while M(F ) andM(F ) are well known as Minkowski content and average Minkowski content of F , re-spectively, provided they are defined.

For self-similar sets F satisfying OSC, it is well known, that the Minkowski dimen-sion coincides with the similarity dimension s of F . Hence sd = s − d, which answerscompletely the question for the right scaling exponent for the volume of the parallel setsλd(Fǫ). However, for a long time it had been an open problem, whether self-similar setsare Minkowski measurable, i.e. whether their Minkowski content exists, although thisquestion aroused considerable interest. After partial answers for sets in R by MichelLapidus and Carl Pomerance [19] and Kenneth Falconer [6], Gatzouras gave the follow-ing classification of Minkowski measurability of self-similar sets in R

d, cf. [11, Theorems2.3 and 2.4].

Theorem 1.2.10. (Gatzouras’s theorem)Let F be a self-similar set satisfying OSC. The average Minkowski content of F alwaysexists and coincides with the strictly positive value

Xd =1

η

∫ 1

0ǫs−d−1Rd(ǫ) dǫ.

If F is non-arithmetic, then also the Minkowski content M(F ) of F exists and equalsXd.

Here the function Rd is the d-th scaling function generalized in the obvious way:

Rd(ǫ) = λd(Fǫ) −N∑

i=1

1(0,ri](ǫ)λd((SiF )ǫ) . (1.2.5)

This theorem contains all the results discussed before for the case k = d and generalizesthem to arbitrary self-similar sets satisfying OSC. Note in particular that the case Xk =0, which causes a lot of trouble in the general discussion, does not occur for k = d, sinceit is possible to show explicitely that always Xd > 0.

With only little extra work we derive in Section 4.6 a proof of Gatzouras’s theoremfrom the proof of Theorem 1.2.6, which differs in some parts from the one provided byGatzouras in [11] and shows more clearly the close connection to curvature measures.Moreover, this proof prepares a strengthening of Gatzouras’s theorem which is presentedin Theorem 1.4.4 below. It characterizes the limiting behaviour of the parallel volumenot only globally but also locally.

19

Fractal Euler numbers. In [22], the fractal Euler number of a set F was introduced as

χf (F ) := limǫ→0

(ǫ

b

)σχ(Fǫ) ,

where b is the diameter of F and σ the Euler exponent (cf. Remark 1.1.3). Similarly,the average fractal Euler number of F was defined as

χf (F ) := limδ→0

1

|log δ|

∫ 1

δ

(ǫ

b

)σχ(Fǫ)

dǫ

ǫ.

The normalizing factor b−σ was inserted to ensure the scaling invariance of these num-bers. It does not affect the limiting behaviour.

We want to work out the relation between Cf0 (F ) and χf (F ) more clearly and com-

pare the results obtained here to those in [22]. Assume that the parallel sets of F are

polyconvex. Then always σ(F ) ≤ s0(F ). If equality holds, then also the numbers Cf0 (F )

and χf (F ) coincide up to the factor b−σ, provided both are defined. The same is true

for their averaged counterparts: χf (F ) = b−σCf0(F ). Therefore, the results obtained for

the 0-th fractal curvatures of self-similar sets can be carried over to the fractal Eulernumbers of these sets. From Theorem 1.2.6 and Corollary 1.2.7 we immediately deducethe following.

Corollary 1.2.11. Let F be a self-similar set satisfying OSC and Fǫ ∈ Rd. If X0 6=0 then σ = s. Moreover, χf (F ) exists and equals the number b−sX0. If F is non-arithmetic, then also χf (F ) = b−sX0.

For the considered class of sets this statement is a significant improvement of the re-sults obtained in [22]. In Corollary 1.2.11, we have no additional assumptions on F apartfrom the OSC. In fact, it can be shown that the additional conditions in Theorem 2.1in [22] are always satisfied in the situation of Corollary 1.2.11. It should be noted thaton the contrary the results in [22] apply to a larger class of self-similar sets. We do notrequire their parallel sets to be polyconvex, since the Euler characteristic is defined moregenerally.

Remark 1.2.12. For sets F in R exactly two fractal curvatures are available, Cf1 (F )

and Cf0 (F ). Since in R for each F and ǫ > 0, Fǫ is a finite union of closed intervals,

Cf1 (F ) is defined if and only if the Minkowski content M(F ) exists and both numbers

coincide. Similarly, since C0(Fǫ, · ) is in this case a positive measure, we always have

s0 = σ and Cf0 (F ) = b−σχf (F ) whenever one of these numbers exists. Corresponding

relations hold for the average counterparts.Fractal Euler numbers in R have been discussed in detail in [22]. Also the close

relation to the gap counting function was outlined there. The limiting behaviour of thegap counting function and the Minkowski content have been studied extensively for setsin R - not only for self-similar sets. We refer in particular to the book by Michel L.Lapidus and Machiel van Frankenhuysen [20]. In this book some kind of Steiner formulawas obtained for general sets F in R, where Minkowski content and gap counting limit,i.e. in fact Cf

0 (F ) and Cf1 (F ), appear as coefficients among others. This suggests that

there are interesting relations between fractal curvatures and the theory of complexdimensions. These connections are still waiting for being studied in detail.

20

Open questions and conjectures. In all the considered examples, in particular in allthe examples presented here and in the succeeding section, we can observe that alwayseither sk = s − k or sk = 0. It is a very interesting question, whether other valuesare possible for sk. We conjecture that this is not the case. For k = d the situation isclear. We always have sd = s − d. Hence sd = 0 only occurs when s = d, i.e. whenthe considered set is full-dimensional like the cubes in Example 1.2.5. For k < d, weconjecture that the case sk = 0 occurs if and only if the set is a ”classical” set, i.e. amongthe class of self-similar sets F satisfying OSC and Fǫ ∈ Rd, exactly those sets have ascaling exponent sk = 0 which are themselves polyconvex. All the other sets in this classhave scaling exponents sk = s − k, and should be regarded as the ”true” fractals. Notethat this classification would be independent of k ∈ 0, . . . , d − 1.

Up to this point we have investigated the limiting behaviour of the total curvaturesCk(Fǫ) for self-similar sets F . Before we continue with the discussion of rescaled limits ofthe curvature measures Ck(Fǫ, · ), we illustrate the obtained results with some examples.

1.3. Examples

We want to illustrate the results of the previous section with some examples and deter-mine the (average) fractal curvatures for some self-similar sets F . To compute the k-thfractal curvature of F , Theorem 1.2.6 and Corollary 1.2.7 suggest to determine first thescaling function Rk and then to compute the number Xk according to formula (1.2.3). If

Xk is different from zero, then we have computed Cfk (F ) (or C

fk(F ), respectively), if not,

we can try to use Theorem 1.2.8. Recall the definition of Rk from (1.2.1) and also for-mula (1.2.2) which will prove very useful for small ǫ. We restrict ourselves to examples in

R2, where we introduced three functionals C

f2(F ), C

f1(F ) and C

f0(F ). Roughly speaking

they can be regarded as fractal volume, fractal boundary length and fractal curvature,

respectively. Recall that Cf2(F ) coincides with the (average) Minkowski content M(F )

and that Cf0(F ) is related to the (average) fractal Euler number.

Example 1.3.1. (Sierpinski gasket)Let F be the Sierpinski gasket generated as usual by three similarities S1, S2 and S3

with contraction ratios 12 such that the diameter of F is 1 and the convex hull M of

F is an equilateral triangle (cf. Figure 1.2). F satisfies the OSC and has polyconvexparallel sets Fǫ. Since F is log 2-arithmetic, the above results ensure only the existenceof average fractal curvatures. (In fact, it can be shown that fractal curvatures do notexist for the Sierpinski gasket.) We determine the scaling functions Rk. It turns outthat they have at most two discontinuities, namely at 1

2 , where the indicator functions

in Rk switch from 0 to 1, and at u =√

312 , the radius of the incircle of the middle triangle

(cf. Figure 1.2), where the intersection structure of the sets (SiF )ǫ changes. For the casek = 0, recall that C0(K) is the Euler characteristic of the set K, i.e. in R

2 the numberof connected components minus the number of ”holes” of K. From Figure 1.3 it is easily

21

u

F

S1M S2M

S3M

M

w1w2

w3

Figure 1.2.: The Sierpinski gasket F and a picture showing how the three similaritiesS1, S2 and S3 generating F act on its convex hull M .

(S1F )ǫ ∩ (S2F )ǫ

(S1F )ǫ ∩ (S3F )ǫ

@@

(S2F )ǫ ∩ (S3F )ǫ

(S1F )ǫ ∩ (S2F )ǫ

(S1F )ǫ ∩ (S3F )ǫ

TTTT

(S2F )ǫ ∩ (S3F )ǫ

Figure 1.3.: Some ǫ-parallel sets of the Sierpinski gasket F for ǫ ≥ u (left) and ǫ < u(middle and right). Note that for ǫ < u the sets (SiF )ǫ ∩ (SjF )ǫ, i 6= j,remain convex and mutually disjoint as ǫ → 0.

22

seen that

R0(ǫ) =

C0(Fǫ) = 1 for 12 ≤ ǫ

C0(Fǫ) −∑

i C0((SiF )ǫ) = −2 for u ≤ ǫ < 12

−∑i6=j C0((SiF )ǫ ∩ (SjF )ǫ) = −3 for ǫ < u. (1.3.1)

Since here s = log 3log 2 and η = log 2, integration according to formula (1.2.3) yields

X0 = −us

ηs= − us

log 3≈ −0.098 .

Being different from zero, X0 is, by Corollary 1.2.7, the value of the 0-th average fractal

curvature Cf0(F ) of F .

For the case k = 1, we use the interpretation that C1(K) is half the boundary lengthof K. Looking at Figure 1.3, it is easily seen that

R1(ǫ) =

C1(Fǫ) = 32 + πǫ for 1

2 ≤ ǫC1(Fǫ) −

∑i C1((SiF )ǫ) = −3

4 − 2πǫ for u ≤ ǫ < 12

−∑i6=j C1((SiF )ǫ ∩ (SjF )ǫ) = −(2π + 3√

3)ǫ for ǫ < u. (1.3.2)

Therefore, by formula (1.2.3),

X1 =1

η

(3

4

us−1

s − 1+ 3

√3us

s

)=

3

4 log 32

us−1 +3√

3

log 3us ≈ 0.82 ,

which is obviously non-zero and thus Cf1(F ) = X1. Similarly for k = 2,

R2(ǫ) =

C2(Fǫ) =√

34 + 3ǫ + πǫ2 for 1

2 ≤ ǫ

C2(Fǫ) −∑

i C2((SiF )ǫ) =√

316 − 3

2ǫ − 2πǫ2 for u ≤ ǫ < 12

−∑i6=j C2((SiF )ǫ ∩ (SjF )ǫ) = −(2π + 3√

3)ǫ2 for ǫ < u

(1.3.3)

and so the average Minkowski content of F (which always exists) is

Cf2(F ) = X2 = −

√3

16 log 34

us−2 +3

2 log 32

us−1 − 3√

3

log 3us ≈ 1.81.

A more explicit formula for Xk. Before we continue with further examples, we providea more convenient formula for Xk, which can reduce the amount of calculation to becarried out. As in the previous example of the Sierpinski gasket the scaling functionsRk are often, though not always, piecewise polynomials of degree at most k. Here wesimply assume such a behaviour for Rk.

Lemma 1.3.2. Let F be a self-similar set with similarity dimension s and polyconvexparallel sets. Let k ∈ 0, . . . , d and, in case s is an integer, assume k < s. Supposethere are numbers J ∈ N and 0 = u0 < u1 < u2 < . . . < uJ < uJ+1 = 1 such that thefunction Rk has a polynomial expansion of degree at most k in the interval (uj , uj+1) foreach j = 0, . . . , J , i.e. there are coefficients aj,l ∈ R such that

Rk(ǫ) =

k∑

l=0

aj,lǫl for ǫ ∈ (uj , uj+1).

23

u

3

2

1

4 5

6

78

Figure 1.4.: Sierpinski carpet Q and a parallel set of Q for ǫ = 19 .

Then, setting aJ+1,l := 0 for each l = 0, . . . , k, it holds

Xk =1

η

k∑

l=0

1

s − k + l

J∑

j=0

(aj,l − aj+1,l)us−k+lj+1 . (1.3.4)

Proof. For a proof of formula (1.3.4), write the integral in formula (1.2.3) as a sum ofintegrals over the intervals (uj+1, uj) and then plug in the polynomial expansions of Rk.

ηXk =

J∑

j=0

∫ uj+1

uj

ǫs−k−1k∑

l=0

aj,lǫldǫ =

J∑

j=0

k∑

l=0

ak,j,l

∫ uj+1

uj

ǫs−k−1+ldǫ

Integration yields 1s−k+l (u

s−k+lj+1 − us−k+l

j ) for the term with indices j and l (providedthat s− k + l 6= 0, which is the case since we assumed s being non-integer or k < s) andso, by exchanging the order of summation,

ηXk =k∑

l=0

1

s − k + l

J∑

j=0

aj,lus−k+lj+1 −

J∑

j=0

aj,lus−k+lj

.

By rearranging the index j in the second sum and summarizing the terms with equal j,formula (1.3.4) easily follows.

Example 1.3.3. (Sierpinski carpet Q)The Sierpinski carpet Q is the well known self-similar set in Figure 1.4 generated by 8similarities Si each mapping the unit square [0, 1]2 with contraction ratio 1

3 to one of

the smaller outer squares. Q has similarity dimension s = log 8log 3 and is log 3-arithmetic.

Thus we can only expect the average fractal curvatures to exist. Indeed, all three ofthem exist and are different from zero as the computations below show.

24

In (0, 1) the scaling functions have discontinuities at 13 , the switching point of the

indicator functions, and 16 , the inradius of the middle cut out square. Between these

points, the intersection structure of the sets Qiǫ := (SiQ)ǫ and the symmetries suggest

to compute Rk as follows:

Rk(ǫ) =

Ck(Qǫ) for 13 ≤ ǫ

Ck(Qǫ) −∑

i Ck(Qiǫ) for 1

6 ≤ ǫ < 13

−∑i6=j Ck(Qiǫ ∩ Qj

ǫ) +∑

i6=j 6=l Ck(Qiǫ ∩ Qj

ǫ ∩ Qlǫ) for ǫ < 1

6

By symmetry arguments, Rk simplifies for ǫ < 16 to

Rk(ǫ) = −8Ck(Q1ǫ ∩ Q2

ǫ ) − 4Ck(Q8ǫ ∩ Q2

ǫ ) + 4Ck(Q8ǫ ∩ Q1

ǫ ∩ Q2ǫ ) .

Now for each scaling function the polynomials for each interval are easily determined(cf. Figure 1.4) and we obtain

R0(ǫ) =

1

−7

−8

, R1(ǫ) =

2 + πǫ

−103 − 7πǫ

−83 − (7π + 4)ǫ

and R2(ǫ) =

1 + 4ǫ + πǫ2

19 − 20

3 ǫ − 7πǫ2

−163 ǫ − (7π + 4)ǫ2

for

13 ≤ ǫ16 ≤ ǫ < 1

3

ǫ < 16

, respectively.

Using formula (1.3.4) we can now compute X0, X1 and X2. Note that η = log 3.

X0 = − 1log 3

1s

(16

)s ≈ −0.0162

X1 = 4log 3

(1

s−1 − 1s

) (16

)s ≈ 0.0725

X2 = 4log 3

(1

s−2 + 2s−1 − 1

s

) (16

)s ≈ 1.352

In the following example we modify the Sierpinski carpet to obtain a self-similar setwith equal dimension but a different geometric and topological structure.

Example 1.3.4. (Modified carpet)The self-similar set M in Figure 1.5 is generated by 8 similarities Si each mapping theunit square with contraction ratio 1

3 to one of the 8 small squares leaving out the uppermiddle one. This time some of the similarities include some rotation by ±π

2 or π as

indicated. Like the Sierpinski carpet, M has similarity dimension s = log 8log 3 and is log 3-

arithmetic. We compute the average fractal curvatures of M and compare them to thoseof the Sierpinski carpet.

In (0, 1) the scaling function R0 has a discontinuity at 13and one at 1

18 , since for ǫ < 118

the first holes appear in M . With similar arguments as for the Sierpinski carpet weobtain that

R0(ǫ) =

1 for 13 ≤ ǫ

−7 for 118 ≤ ǫ < 1

3

−14 for ǫ < 118

,

25

Figure 1.5.: Modified Sierpinski carpet M and some parallel sets

and so integration according to formula (1.2.3) yields

Cf0(M) = X0 = − 1

s log 3

7

8

(1

6

)s

≈ −0.014 .

For the cases k = 1 and k = 2, the situation is more complicated. First observe that

C1(Fǫ) =

116 + (π + arcsin 1

6ǫ)ǫ for 16 ≤ ǫ

73 + (3

2π − 2)ǫ for 118 ≤ ǫ < 1

6

and similarly for i = 1, . . . , 8,

C1((SiM)ǫ) =11

18+

(π + arcsin

1

18ǫ

)ǫ for

1

18≤ ǫ .

From these two equations R1(ǫ) can be determined in the interval [ 118 , 1) by means of

the relationR1(ǫ) = C1(Fǫ) − 8 C1((SiM)ǫ) 1(0, 1

3](ǫ) .

Obviously, this time R1 is not piecewise a polynomial as in the previous examples. Forǫ < 1

18 , we derive R1 from the intersections of the (SiM)ǫ and obtain

R1(ǫ) = −20

9− 7

(3

2π + 2

)ǫ .

Integrating R1 according to (1.2.3) yields

Cf1(M) = X1 ≈ 0.0720 .

Similarly, for k = 2 we determine the area of Fǫ:

C2(Fǫ) =

1 + 11

3 ǫ + (π + arcsin 16ǫ)ǫ

2 + 16

√ǫ2 −

(16

)2for 1

6 ≤ ǫ89 + 14

3 ǫ + (32π − 2)ǫ2 for 1

18 ≤ ǫ < 16

26

and for i = 1, . . . , 8,

C2((SiM)ǫ) =1

9+

11

9ǫ +

(π + arcsin

1

18ǫ

)ǫ2 +

1

18

√

ǫ2 −(

1

18

)2

for1

18≤ ǫ .

Now R2(ǫ) can be derived for ǫ ∈ [ 118 , 1) using that

R2(ǫ) = C2(Fǫ) − 8 C2((SiM)ǫ) 1(0, 13](ǫ).

For ǫ < 118 , we look again at the intersections of the (SiM)ǫ and obtain

R2(ǫ) = −40

9ǫ − 7

(3

2π + 2

)ǫ2.

Integrating R2 according to (1.2.3) yields

Cf2(M) = X2 ≈ 1.3439 .

Comparison of the carpets. We summarize the approximate values determined abovefor the fractal curvatures of the two carpets:

Cf0 Cf

1 Cf2

Sierpinski carpet −0.016 0.0725 1.352

Modified carpet −0.014 0.0720 1.344

The corresponding fractal curvatures are different. Hence they can be used to distinguishboth sets. On the other hand the values are rather close to each other, which correspondsto the impression that geometrically the sets are not very different. More investigationsare necessary to understand, whether fractal curvature are useful characteristics for thedistinction and classification of fractal sets, and whether they have some more explicitgeometric interpretation. In particular it would be interesting to see, whether sets withvery similar geometric structure also have fractal curvatures which are very close to eachother, i.e. whether there is some kind of continuity.

In contrast to the cube Q in Example 1.2.5 it can happen, that for a self-similar setXk equals zero but nevertheless sk = s−k. The next example is a set, for which X0 = 0but s0 = s. It also clarifies the difference between s0 and the Euler exponent σ, definedin [22].

Example 1.3.5. (Sierpinski tree)Let the set F be generated by three similarities S1, S2 and S3. S1 has contraction ratio45 and shrinks an equilateral triangle of diameter 1 towards one of its corners, whilethe other two similarities S2 and S3 map the triangle with ratio 1

5 to the remainingtwo corners, including a rotation by 2π

3 and −2π3 , respectively (cf. Figure 1.6). F has

similarity dimension s given by 5s − 4s = 2 and is non-arithmetic, since log 5−log 4log 5 is not

rational. Therefore, this time we can try to determine the fractal curvatures rather than

27

4

5

1

5

S1M

S2M S3M

M

Figure 1.6.: The Sierpinski tree F and how it is generated. The right picture indicateshow the similarities generating F act on its convex hull M . The arrowsindicate to which points the upper corner of M is mapped.

x

B(x, ǫ0)

ǫ0O

−ǫ0

B(x, ǫ0)

x

A

O

Figure 1.7.: Feasible open set O of the Sierpinski tree F and some inner parallel setO−ǫ0 . The enlarged part on the right shows some detail of the ǫ-parallel setof F near x for some ǫ < ǫ0. The arc A ⊂ ∂Fǫ contributes 1

3 to the 0-thcurvature of Fǫ.

28

just the averaged counterparts. We compute Cf0 (F ), for which we first determine the

scaling function R0. The only discontinuities of R0(ǫ) in (0, 1) are 45 and 1

5 and so

R0(ǫ) =

C0(Fǫ) = 1 for 45 ≤ ǫ

C0(Fǫ) −∑

i C0((SiF )ǫ) = 0 for 15 ≤ ǫ < 4

5

−C0((S1F )ǫ ∩ (S2F )ǫ) − C0((S1F )ǫ ∩ (S3F )ǫ) = −2 for ǫ < 15

.

By formula (1.3.4),

X0 =1

ηs

(1 −

(4

5

)s

− 2

(1

5

)s)= 0 .

Unfortunately, Corollary 1.2.7 does not allow to conclude now directly that s0 = s andthus Cf

0 (F ) = 0. But this is in fact true and will be derived from Theorem 1.2.8. Letx and O be as indicated in Figure 1.6. Note that O is a feasible open set of F . Choosesome ǫ0 < 1

20 and let B = B(x, ǫ0) the ball with center x and radius ǫ0. It is not difficultto see that B ⊂ O−ǫ0 , since d(x, ∂O) = 1

10 . Moreover, for ǫ ≤ ǫ0, Cvar0 (Fǫ, B) ≥ 1

3 , sincethe set ∂Fǫ ∩ B does always contain the arc A whose length is 1

3 of the perimeter ofthe circle with center x and radius ǫ. This contributes the amount of 1

3 to the mass ofC+

0 (Fǫ, B) and thus to Cvar0 (Fǫ, B). Now Theorem 1.2.8 implies that ǫsCvar

0 (Fǫ) ≥ c for

ǫ < ǫ0 (where c = 13

(ǫ05

)s) and so, by Corollary 1.2.9, s0 = s. Hence Cf

0 (F ) = 0 asclaimed.

The above example contrasts Example 1.2.5, where we also had X0 = 0 but s0 < s.While for the parallel sets of the cube Q not only the 0-th total curvature C0(Qǫ) remainsbounded as ǫ → 0 but also the local 0-th curvature, here locally the curvature grows asǫ → 0 and only the total curvature C0(Fǫ) remains bounded (in fact constant). For allǫ, the positive curvature of Fǫ equals out its negative curvature. In [22] we consideredthe Euler characteristic, which equals to the 0-th total curvature. It does not ”see” thelocal behaviour of the curvature and therefore we obtain σ = 0 and χf (F ) = 1, whichreflects somehow its topolocical structure (connected and simply connected) but not its

”fractality” which is better revealed from s0 = s and Cf0 (F ) = 0 (0-th curvature scales

locally with ǫs but vanishes globally).

Remark 1.3.6. In Remark 1.1.4 we introduced scaling exponents s+k and s−k , by taking

in Definition 1.1.2 only the positive or the negative curvature, respectively, into account.In our examples in R

2 this distinction does only make sense for k = 0. It is not difficultto see, that for the Sierpinski gasket or the carpets only the negative curvature C−

0 (Fǫ)increases as ǫ → 0, while the positive curvature C+

0 (Fǫ) remains bounded. Therefore,s−0 = s0 = s and s+

0 = 0 < s0 in those examples. The situation is different for theSierpinski tree. Here the negative and positive curvature grow with the same speed.Hence s+

0 = s−0 = s0 = s.

1.4. Fractal curvature measures

With the convergence of rescaled total curvatures as discussed in Section 1.2 naturallythe question arises how the corresponding (rescaled) curvature measures behave. It turns

29

out that, for self-similar sets F , typically weak limits of rescaled curvature measures existtoo. These limit measures should be regarded as the fractal curvature measures of F . Weshow that they are, in fact, all multiples of the same measure, namely the s-dimensionalHausdorff measure on F .

Rescaled curvature measures. As before, let F ⊂ Rd be a compact set such that for

all ǫ > 0, Fǫ ∈ Rd. This ensures that the curvature measures Ck(Fǫ, · ) are defined foreach ǫ. We want to study the limiting behaviour of these measures as ǫ → 0. Thisis motivated by the question, whether it is possible to define some kind of curvaturemeasures for fractals in this way. The weak convergence of signed measures seems tobe the appropriate notion of convergence here. It is the straightforward generalizationof the usual weak convergence of positive measures. For each ǫ > 0, let µǫ be a totallyfinite signed measure on R

d. The measures µǫ are said to converge weakly to a totallyfinite signed measure µ as ǫ → 0, µǫ

w−→ µ, if and only if∫

Rd fdµǫ →∫

Rd fdµ for allbounded continuous functions f on R

d. We refer to the Appendix for more details.Since weak convergence is always accompanied by the convergence of the total masses

of the measures, it is clear that the measures Ck(Fǫ, · ) have to be rescaled with thefactor ǫsk , where sk is the k-th scaling exponent as defined above. Therefore we makethe following definitions: For each ǫ > 0 and k = 0, . . . , d we define the k-th rescaledcurvature measure νk,ǫ of Fǫ by

νk,ǫ( · ) := ǫskCk(Fǫ, · ). (1.4.1)

In general, these measures need not converge weakly as ǫ → 0. We have seen abovethat often the total mass νk,ǫ(R

d) = ǫskCk(Fǫ) already fails to converge, which makesweak convergence impossible. Therefore, in analogy with Definition 1.1.6, we also defineaveraged versions νk,ǫ of the rescaled curvature measures νk,ǫ by

νk,ǫ( · ) :=1

|log ǫ|

∫ 1

ǫǫskCk(Fǫ, · )

dǫ

ǫ. (1.4.2)

For k = d and k = d − 1, the measure Ck(Fǫ, · ) is known to be positive, hence so areνk,ǫ and νk,ǫ. In general, the measures νk,ǫ and νk,ǫ are totally finite signed measures.

Weak convergence for self-similar sets. Let F be a self-similar set satisfying OSC.Denote by µF the normalized s-dimensional Hausdorff measure on F , i.e.

µF =Hs|F ( · )Hs(F )

. (1.4.3)

It is well known that the OSC implies 0 < Hs(F ) < ∞. Hence µF is well defined. Asbefore, we additionally assume that the parallel sets of F are polyconvex.

We ask for the weak convergence of the rescaled curvature measures and their averagedversions and restrict this question to sets for which sk = s − k, since for sets withsk < s − k we do not even have information on the convergence behaviour of the totalmasses of these measures. It is again necessary to distinguish between arithmetic andnon-arithmetic self-similar sets F . The weak convergence of the measures νk,ǫ as ǫ → 0

30

is only ensured in the non-arithmetic case, while the measures νk,ǫ converge in general.This is not very surprising, since the convergence of the total masses νk,ǫ(R

d) was ensuredby Theorem 1.2.6 only for non-arithmetic sets, while the total masses νk,ǫ(R

d) convergein general. So the value of the total mass of the limit measure (if it exists) must be

Cfk (F ) or C

fk(F ), respectively. The following statement gives an answer to when weak

limits exist and what the limit measures are.

Theorem 1.4.1. Let F be a self-similar set satisfying OSC and Fǫ ∈ Rd, k ∈ 0, . . . , d.Assume sk = s − k. Then always

νk,ǫw−→ C

fk(F ) µF as ǫ → 0 .

If − log r1, . . . ,− log rN is non-arithmetic, then

νk,ǫw−→ Cf

k (F ) µF as ǫ → 0 .

For each k, the limit measure is some multiple of the measure µF . Note that the case

Cfk(F ) = 0 is included in this formulation. For this case the limit is the zero measure.

Otherwise it is either a positve or a purely negative measure, depending on the signum

of the factor Cfk (F ) or C

fk(F ), respectively. The limit measure Cf

k (F ) µF (or Cfk(F ) µF ,

respectively) should be regarded as the k-th fractal curvature measure of F . The theoremstates that the d + 1 fractal curvature measures of F all coincide up to some constantfactors. Taking into account the self-similarity of the considered sets, it is not verysurprising that all fractal curvature measures essentially coincide with µF . Any measureon a self-similar set F describing its geometry should respect the self-similar structureof F . But the self-similar measures on F are well known and so it is not surprising toretrieve them here. The proof of this theorem will be provided in Section 5.2.

Weak limits of the parallel volume. For k = d, Theorem 1.4.1 can be generalizedto arbitrary self-similar sets satisfying OSC. By replacing Cd(Fǫ, · ) with the Lebesguemeasure, we can again drop the assumption of polyconvexity for the parallel sets Fǫ.The definition of the rescaled measure νd,ǫ in (1.4.1) generalizes to arbitrary compactsets F ⊂ R

d by settingνd,ǫ( · ) := ǫsdλd(Fǫ ∩ · ). (1.4.4)

Similarly, (1.4.2) generalizes to

νd,ǫ( · ) :=1

|log ǫ|

∫ 1

ǫǫsdλd(Fǫ ∩ · )dǫ

ǫ. (1.4.5)

We call νd,ǫ the rescaled ǫ-parallel volume and νd,ǫ the average rescaled ǫ-parallel volumeof F , respectively. If some weak limit of these measures exist, as ǫ → 0, then thetotal mass of the limit measure must coincide with the Minkowski content M(F ) orits average counterpart M(F ), respectively. Indeed, such limit measures exist as thefollowing statement shows.

Theorem 1.4.2. Let F be a self-similar set satisfying OSC. Then always

νd,ǫw−→ M(F ) µF as ǫ → 0 .

31

If F is non-arithmetic, then also

νd,ǫw−→ M(F ) µF as ǫ → 0 .

This result extends Gatzouras’s theorem. Not only the total (average) ǫ-parallel vol-ume of self-similar sets converges, as ǫ → 0. The convergence happens even locally inevery ”nice” subset of R

d. This is the meaning of weak convergence. More precisely, ifB ⊂ R

d is a µF -continuity set, i.e. if µF (∂B) = 0, then νd,ǫ(B) → M(F ) µF (B) as ǫ → 0for F non-arithmetic and νd,ǫ(B) → M(F )µF (B) correspondingly for general F .

Normalized curvature measures We also want to explore another type of limit forthe curvature measures which avoids the averaging and yields convergence nevertheless.Although in our situation averaging is a natural procedure to improve the convergencebehaviour, it is not the only possible one. Another way to overcome the problem ofoscillations, which prevent the convergence, is to normalize the measures. Since normal-ization is only possible for positive and finite measures and since the rescaled curvaturemeasures νk,ǫ are in general signed measures for k ∈ 0, . . . , d − 2, this does only makesense for k = d − 1 and k = d. We discuss both cases separately, since for k = d we canagain obtain more general results.

The case k = d − 1. Define the d − 1-th normalized curvature measure of Fǫ by

ν1d−1,ǫ( · ) :=

νd−1,ǫ( · )νd−1,ǫ(Rd)

=Cd−1(Fǫ, · )Cd−1(Fǫ)

.

Theorem 1.4.3. Let F be a self-similar set satisfying OSC and Fǫ ∈ Rd. Assume thatXd−1 := lim infǫ→0 ǫs−d+1Cd−1(Fǫ) > 0. Then

ν1d−1,ǫ

w−→ µF as ǫ → 0 .

Observe that the measures ν1d−1,ǫ converge weakly even in the arithmetic case. No

distinction is necessary between arithmetic and non-arithmetic self-similar sets. Thenormalization has a similar effect as the averaging. The additional assumption Xd−1 > 0implies that in particular Xd−1 > 0, since Xd−1 ≥ Xd−1. Hence sd−1 = s− d+1. In thenon-arithmetic case this assumption is equivalent to Xd−1 > 0. For arithmetic sets it isslightly stronger. The proof of Theorem 1.4.3 is given in Section 5.3.

In general, this result does not carry over to the non-normalized counterparts νd−1,ǫ.Under the conditons of Theorem 1.4.3, we obviously have the relation

νd−1,ǫ = ǫsd−1Cd−1(Fǫ) ν1d−1,ǫ.

In the arithmetic case, the convergence of the prefactor ǫsd−1Cd−1(Fǫ) is not assured andso in general the existence of a weak limit of these measures can not be derived fromthis result.

32

The case k = d. Define the normalized parallel volume νd,ǫ of Fǫ by

ν1d,ǫ( · ) =

νd,ǫ( · )νd,ǫ(Rd)

=λd(Fǫ ∩ · )

λd(Fǫ).

The measures νd,ǫ are well defined for each compact set F ⊆ Rd and ǫ > 0. For self-

similar sets F satisfying OSC the normalized parallel volume converges weakly to µF asthe following Theorem states.

Theorem 1.4.4. Let F be a self-similar set satisfying OSC and Fǫ ∈ Rd. Then

ν1d,ǫ

w−→ µF as ǫ → 0 .

Again it is not necessary to distinguish between arithmetic and non-arithmetic self-similar sets. Here no additional assumptions are required, since, by Gatzouras’s theorem,always Xd > 0. The proof of Theorem 1.4.4 can be found in Section 5.3.

33

2. Curvature measures in the convex ring

Curvature measures were introduced by Herbert Federer in 1959 in his famous paperCurvature measures [8] for sets with positive reach by means of a local Steiner formula.A set K ⊆ R

d is said to have positive reach if there exist some ǫ > 0 such that eachpoint x ∈ Kǫ has a unique nearest point in K. The reach of K is the supremum overall such ǫ. Convex sets are special sets with positive reach, namely those with infinitereach and are thus included in his theory. For convex sets many additional results havebeen established, for instance an axiomatic characterization of curvature measures byRolf Schneider [28].

The total curvatures, also known as intrinsic volumes or Minkowski functionals, hadbeen studied long before by Herbert Minkowski and others. Much of the theory wascollected and developed further by Hugo Hadwiger in his famous book [13]. A milestonewas to interpret intrinsic volumes as valuations and use their additivity to extend themto the convex ring. The corresponding additive extension of curvature measures topolyconvex sets is due to Helmut Groemer [12], while a more explicit characterization ofcurvature measures in the convex ring was given by Rolf Schneider [29], who generalizedthe local Steiner formula by weighting the parallel volume with an index function.

There has also been considerable progress in the development of curvature measuresfor non-convex (and non-polyconvex) sets since the time of Federer, which has not yetcome to an end. On this subject, which is not in the scope of the theory required here,we refer to the survey paper of Andreas Bernig [2] and the references given therein.

Here we want to collect some facts about curvature measures in the convex ring.In particular, we outline the most important properties of curvature measures and theirvariation measures. For the latter we also derive some useful estimates, which we requirelater on. Our main references for the mostly well known statements in this chapter arethe books of Rolf Scheider [30] and Rolf Scheider and Wolfgang Weil [31]. Also comparethe monograph by Daniel A. Klain and Gian-Carlo Rota [17].

2.1. Properties of curvature measures

Local Steiner formula. For convex bodies K, curvature measures are usually intro-duced as the coefficients in the polynomial expansion of the local parallel volume of K.The metric projection πK onto a convex set K ∈ Kd maps each point x ∈ R

d to its(uniquely determined) nearest neighbour in K. For fixed K ∈ Kd and ǫ > 0 the setKǫ ∩ π−1

K (B) is regarded as the local parallel set of K with respect to the Borel set B.It consists of the two sets K ∩ B and x ∈ Kǫ \ K : πK(x) ∈ B. The Steiner formuladescribes the volume of the local parallel set of K as a polynomial in ǫ. Let κk denotethe k-dimensional volume of the unit ball in R

k and Hd the d-dimensional Hausdorffmeasure.

35

Theorem 2.1.1. For each K ∈ Kd there exist uniquely determined finite Borel measuresC0(K, · ), . . . , Cd(K, · ) on R

d, called the curvature measures of K, such that

Hd(Kǫ ∩ π−1K (B)) =

d∑

k=0

ǫkκkCd−k(K, B)

for each Borel set B and ǫ > 0.

The proof is done in two steps. By decomposing polytopes into their faces, explicitexpressions for their curvature measures and parallel volume can be derived. The generalcase follows by approximation of convex sets with polytopes.

Additivity. Curvature measures have the property of being additive. If K, L ∈ Kd suchthat also K ∪ L ∈ Kd, then

Ck(K ∪ L, B) = Ck(K, B) + Ck(L, B) − Ck(K ∩ L, B) (2.1.1)

for each Borel set B ⊆ Rd. Repeated application of this relation leads to the so called

inclusion-exclusion principle: If K1, . . . , Km and K =⋃m

i=1 Ki are in Kd, then for allBorel sets B ⊆ R

d

Ck(K, B) =∑

I∈Nm

(−1)#I−1Ck

(⋂

i∈I

Ki, B). (2.1.2)

Here Nm denotes the family of all nonempty subsets I of 1, . . . , m, so that the sumruns through all intersections of the Ki.

Further properties. Curvature measures of convex bodies have several further proper-ties which we summarize now. Let K, L denote convex bodies in Kd and B ⊆ R

d anarbitrary Borel set. For each k ∈ 0, . . . , d it holds:

Motion covariance: If g is an Euclidean motion, then Ck(gK, gB) = Ck(K, B).

Homogeneity of degree k: For λ > 0, Ck(λK, λB) = λkCk(K, B).

Locality: If K ∩A = L∩A for some open set A ⊆ Rd, then Ck(K, B) = Ck(L, B) for all

Borel sets B ⊆ A.

Continuity in the first argument: If K1, K2, . . . ∈ Kd such that Ki → K (in the Haus-dorff metric) as i → ∞ then Ck(K

i, · ) w−→ Ck(K, · ). In particular, Ck(Ki) → Ck(K).

Monotonicity of the total curvature: If K, L ∈ Kd and K ⊆ L, then Ck(K) ≤ Ck(L).

The above properties allow to characterize curvature measures axiomatically. Ac-cording to Schneider [28], any continuous, additive and motion covariant functional Φassigning to each convex body K ∈ Kd a finite measure Φ(K, · ) is a linear combinationof the curvature measures Ck, i.e. there exist constants c0, . . . , cd such that Φ =

∑k ckCk.

Demanding additionally homogeneity of degree k from Φ is sufficient to characterize itas a constant multiple of the k-th curvature measure.

36

Additive extension to the convex ring. In case the set K on the left hand side of(2.1.2) is not convex, the measure Ck(K, · ) is not defined by the local Steiner formula.But then the right hand side could be regarded as its definition. This leads to theadditive extension of curvature measures to the convex ring. Helmut Groemer showedin [12] that this extension is indeed possible, i.e. the so defined measures Ck(K, · ) donot depend on the chosen representation of K by convex sets Ki.

Theorem 2.1.2. For each k ∈ 0, . . . , d, the k-th curvature measure Ck has a uniqueadditive extension to Rd.

Curvature measures of polyconvex sets are in general signed measures, in contrastto the convex case. However, for k = d and d − 1, Ck(K, · ) is a positive measure foreach K ∈ Rd. Most of the properties of curvature measures of convex sets generalize topolyconvex sets. By definition, they are additive and so the inclusion-exclusion principle(2.1.2) is valid for all K, Ki ∈ Rd. Moreover, for each k = 0, . . . , d the k-th curvaturemeasure is motion covariant, homogeneous of degree k and has the locality property.Continuity and Monotonicity property do not generalize to polyconvex sets, which iseasily seen from simple counter-examples.

Since each similarity S is the composition of an Euclidean motion and a homothetywith some ratio r > 0, the k-th curvature measure has the following scaling propertywith respect to S:

Ck(SK, SB) = rkCk(K, B) (2.1.3)

for K ∈ Rd and any Borel set B ∈ Rd.

Curvature of parallel sets. Since any parallel set of a convex set is again convex, theparallel sets of polyconvex sets are polyconvex as well (compare Fact 1.1.1) and theircurvature measures are defined. For sets K ∈ Rd we are particularly interested in thecontinuity properties of the total curvatures Ck(Kǫ) as a function of ǫ. The statementbelow is a consequence of the weak continuity of curvature measures for convex sets.

Lemma 2.1.3. For K ∈ Rd and k ∈ 0, . . . , d, Ck(Kǫ), as a function of ǫ, has a finiteset of discontinuities in (0,∞) and limǫ→0 Ck(Kǫ) = Ck(K). (For k = d, Ck(Kǫ) is evencontinuous in (0,∞).)

Proof. Let K1, . . . , Km ∈ Kd be sets such that K =⋃m

i=1 Ki. Then by the inclusion-exclusion principle,

Ck(Kǫ) =∑

I∈Nm

(−1)#I−1Ck

(⋂

i∈I

Kiǫ

), (2.1.4)

where the sets⋂

i∈I Kiǫ are convex (possibly empty) for all ǫ ≥ 0. (Here K0 = K.) More

precisely, for each I there exists ǫI ≥ 0 such that⋂

i∈I Kiǫ = ∅ for all 0 ≤ ǫ < ǫI and⋂

i∈I Kiǫ 6= ∅ for all ǫ ≥ ǫI . Now the continuity property implies that Ck(

⋂i∈I Ki

ǫ) iscontinuous in (ǫI ,∞) and continuous from the right in ǫI . Moreover, Ck(

⋂i∈I Ki

ǫ) ≡ 0in [0, ǫI) and thus the only possible discontinuity point in (0,∞) is ǫI . Since this holdsfor every I ∈ Nm, by (2.1.4), Ck(Kǫ) has finitely many discontinuities in (0,∞) (at most#Nm). In particular, since always Ck(

⋂i∈I Ki

ǫ) → Ck(⋂

i∈I Ki) as ǫ → 0, we concludethat Ck(Kǫ) → Ck(K).

37

2.2. Variation measures

Some of the above properties of curvature measures carry over to the correspondingvariation measures. Recalling that the positive, negative and total variation of themeasure Ck(K, · ) are given by

C+k (K, B) = sup

B′⊆BCk(K, B′), C−

k (K, B) = − infB′⊆B

Ck(K, B′)

and Cvark (K, B) = C+

k (K, B) + C−k (K, B) respectively, for each Borel set B ⊆ R

d, thefollowing is easily seen.

Proposition 2.2.1. For k ∈ 0, . . . , d and K ∈ Rd the measures C+k (K, · ), C−

k (K, · )and Cvar

k (K, · ) are motion covariant, homogeneous of degree k and have the localityproperty.

Proof. Let g a Euclidean motion. The motion covariance of C+k (K, · ) follows from the

same property of Ck(K, · ) observing that for all Borel sets B

C+k (gK, gB) = sup

gB′⊆gBCk(gK, gB′) = sup

B′⊆BCk(K, B′) = C+

k (K, B).

The remaining properties are verified in a similar way, as well for C−k (K, · ). Then,

for Cvark (K, · ), they follow immediately from the relation Cvar

k (K, B) = C+k (K, B) +

C−k (K, B).

Note that also the scaling property remains valid for the variation measures: If k ∈0, . . . , d, • ∈ +,−, var and S is a similarity, then

C•k(SK, SB) = rkC•

k(K, B) (2.2.1)

for each K ∈ Rd and any Borel set B ⊆ Rd.

It would be very useful also to have a generalization of Lemma 2.1.3 to the variationmeasures of a curvature measure. Unfortunately, we have no idea how to prove this.Nevertheless, we believe that the following statement is true.

Conjecture 2.2.2. For K ∈ Rd, k ∈ 0, . . . , d − 2 and • ∈ +,−, var, C•k(Kǫ), as a

function of ǫ, has a finite set of discontinuities in (0,∞) and limǫ→0 Ck(Kǫ) = Ck(K).

Since for k = d − 1 and k = d, Ck(K, · ) is a positive measure, the situation is clearfor those cases. A verification of this statement would lead to some extensions of themain results to the variation measures and simplify the proofs in Chapter 5. For moredetails confer Remark 4.5.2.

Estimates for the total variation measure. Above we derived some properties of thevariation measures directly from the corresponding properties of the underlying cur-vature measure. Unfortunately, additivity is not among them. Indeed, the variationmeasures of a curvature measure fail to be additive in general. Therefore, we now derivesome inequalities for the total variation measures, which, in a way, take over the rolethe inclusion-exclusion principle plays for curvature measures.

38

Let Kj ∈ Rd for j = 1, . . . , m and K =⋃m

j=1 Kj . Recall that Nm was the family

of all nonempty subsets of 1, . . . , m. For each I ∈ Nm write K(I) :=⋂

j∈I Kj .The additivity of curvature measures allows to derive some estimates for their variationmeasures.

Lemma 2.2.3. Let Kj ∈ Rd for j = 1, . . . , m and K =⋃m

j=1 Kj. Then

Cvark (K, B) ≤

∑

I∈Nm

Cvark (K(I), B) (2.2.2)

for each Borel set B. If the sets Kj are convex then

Cvark (K, B) ≤ (2m − 1)max

jCk(K

j). (2.2.3)

Note that, since C±k (K, · ) ≤ Cvar

k (K, · ), estimate (2.2.3) remains valid if Cvark (K, B)

on the left hand side is replaced with C+k (K, B) or C−

k (K, B). In general, inequality(2.2.2) does not remain valid when the total variation is replaced with the positive ornegative variation measure.

Proof. Let Rd = K+∪K− be a Hahn decomposition of the signed measure Ck(K, · ), i.e.

K+ and K− are disjoint sets satisfying C+k (K, K−) = C−

k (K, K+) = 0 (cf. Appendix,Theorem A.1.1). By the inclusion-exclusion formula, it holds

C±k (K, B) = (±1)Ck(K, B ∩ K±) = (±1)

∑

I∈Nm

(−1)#I−1Ck(K(I), B ∩ K±),

and, since |Ck(K, · )| ≤ Cvark (K, · ),

C±k (K, B) ≤

∑

I∈Nm

Cvark (K(I), B ∩ K±).

Hence we obtain

Cvark (K, B) = C+

k (K, B) + C−k (K, B)

≤∑

I∈Nm

Cvark (K(I), B ∩ K+) +

∑

I∈Nm

Cvark (K(I), B ∩ K−)

=∑

I∈Nm

Cvark (K(I), B)

for each Borel set B ⊆ Rd, as stated in (2.2.2). The second assertion follows from the

first one by noting that for each I ∈ Nm, the set K(I) is convex and thus Cvark (K(I), · ) =

Ck(K(I), · ). Since K(I) ⊆ Kj for some j ∈ 1, . . . , m, the monotonicity of the totalcurvatures in Kd yields Ck(K(I), B) ≤ Ck(K(I)) ≤ maxj Ck(K

j). Observing now thatthe number of summands, i.e. the number of sets in Nm, is 2m − 1, the second assertionfollows.

39

The above estimates are not satisfactory in case we have a large number m of setsKj with comparably few mutual intersections. But in this situation it can be improvedeasily. Define the intersection number Γ = Γ(X ) of a finite family X =

K1, . . . , Km

of sets as the maximum over all l ∈ 1, . . . , m of the number of nonempty intersectionsK l ∩ Kj with Kj ∈ X , i.e.

Γ(X ) = maxl

#

j : Kj ∩ K l 6= ∅

. (2.2.4)

If Γ is small compared to m, then the following estimate is useful.

Corollary 2.2.4. Let Kj ∈ Rd for j = 1, . . . , m and K =⋃m

j=1 Kj, Γ the intersection

number of the familyK1, . . . , Km

and b > 0 such that for all I ∈ Nm

Cvark (K(I), B) ≤ b.

Then

Cvark (K, B) ≤ m2Γb.

Proof. Let Il be the set of all indices j such that K l ∩Kj 6= ∅. For each K l it suffices toconsider its intersections with sets Kj with j ∈ Il, all other intersections being empty.Therefore the sum on the right hand side of (2.2.2) is contained in

m∑

l=1

∑

I⊆Il

Cvark (K l ∩ K(I), B),

where we set K(I) := Rd in case I = ∅. By assumption, each term is bounded from

above by b. Moreover, #Il ≤ Γ and so the number of subsets of Il is not greater than2Γ. Hence the asserted estimate follows.

Now assume that L =⋂m

j=1 Kj is the intersection of a finite number of sets Kj ∈ Rd.Again we ask for an upper bound of Cvar

k (L, · ) in terms of the curvatures of the setsKj . For convex sets Kj there is an obvious bound for the total curvature. In this casethe set L is convex as well and the monotonicity implies

Ck(L) ≤ minj=1,...,m

Ck(Kj).

If the Kj are polyconvex, we have at least some bound in terms of representations ofKj with convex sets.

Lemma 2.2.5. Let L =⋂m

j=1 Kj. Assume that each Kj has a representation Kj =⋃P

i=1 Kj,i as a union of (at most) P convex sets Kj,i. Then

Cvark (L) ≤ (2(P m) − 1)max

j,iCk(K

j,i).

Note that, since C±k (L, · ) ≤ Cvar

k (L, · ), the assertion also holds in case Cvark (L) is

replaced with C+k (L) or C−

k (L).

40

Proof. We have

L =

m⋂

j=1

Kj =

m⋂

j=1

P⋃

ij=1

Kj,ij =

P⋃

i1,...,im=1

m⋂

j=1

Kj,ij

,

i.e. L is the union of the Pm convex sets⋂

j Kj,ij . Therefore, the assertion followsimmediately from the second statement in Lemma 2.2.3.

41

3. Adapted Renewal theorem

For the proofs of Theorem 1.2.6 and Theorem 1.2.10 we require the Renewal theoremwhich we recall and discuss now. Afterwards we will reformulate it in a way which ismost convenient for our purposes. Later on we will only use this variant of the Renewaltheorem. It is stated in Theorem 3.1.4.

The Renewal theorem. Let P be a Borel probability measure with support containedin [0,∞) and η :=

∫∞0 tP (dt) < ∞. Let z : R → R be a function with a discrete set of

discontinuities satisfying

|z(t)| ≤ c1e−c2|t| for all t ∈ R (3.1.1)

for some constants 0 < c1, c2 < ∞. It is well known in probability theory that underthese conditions on z the equation

Z(t) = z(t) +

∫ ∞

0Z(t − τ)P (dτ) (3.1.2)

has a unique solution Z(t) in the class of functions satisfying limt→−∞ Z(t) = 0. Equa-tion (3.1.2) is called a renewal equation and the asymptotic behaviour of its solution ast → ∞ is given by the so-called Renewal theorem. The Renewal theorem is a standardtool in probability theory (cf. e.g. Feller [9]). In the last years, it has been discoveredas a tool in fractal geometry too. Therefore, versions are available which are adapted tothe fractal setting. In fractal applications, P usually is a probability measure supportedby a finite set of points y1, . . . , yN ∈ [0,∞) such that P (yi) = pi for i = 1, . . . , N andtherefore

η =N∑

i=1

yipi. (3.1.3)

Discrete versions of the Renewal theorem are for instance provided in Kenneth Falconer’sbook [7, Corollary 7.3, p. 122]) or in the paper [21] of Michael Levitin and DmitriVassiliev.

A function g : R → R is said to be asymptotic to a function f : R → R, g ∼ f , if forall ǫ > 0 there exists a number D = D(ǫ) such that

(1 − ǫ)f(t) ≤ g(t) ≤ (1 + ǫ)f(t) for all t > D. (3.1.4)

Recall from Section 1.2 that the set y1, ..., yN is called h-arithmetic if h is the largestnumber such that yi ∈ hZ for i = 1, . . . , N and non-arithmetic if no such number hexists.

43

Theorem 3.1.1. (Renewal theorem) Let 0 < y1 ≤ y2 ≤ . . . ≤ yN and p1, . . . , pN bepositive real numbers such that

∑Ni=1 pi = 1. For a function z as defined in (3.1.1), let

Z : R → R be the unique solution of the renewal equation

Z(t) = z(t) +N∑

i=1

piZ(t − yi) (3.1.5)

satisfying limt→−∞ Z(t) = 0. Then the following holds:

(i) If the set y1, ..., yN is non-arithmetic, then

limt→∞

Z(t) =1

η

∫ ∞

−∞z(τ)dτ.

(ii) If y1, ..., yN is h-arithmetic for some h > 0, then

Z(t) ∼ h

η

∞∑

k=−∞z(t − kh).

Moreover, Z is uniformly bounded in R.

Theorem 3.1.1 implies that in the non-arithmetic case the limit limt→∞ Z(t) exists,while in the h-arithmetic case Z is asymptotic to some periodic function of period h > 0(i.e. to some function f with f(t+h) = f(t) for all t ∈ R). The latter is sufficient for the

limit limT→∞1T

∫ T0 Z(t)dt to exist which is easily derived from the following observation.

Lemma 3.1.2. Let f be a locally integrable periodic function with period h > 0 and letL :=

∫ h0 f(t)dt.

(i) Then the limit limT→∞

1T

∫ T0 f(t)dt exists and equals h−1L.

(ii) If g : R → R is a function such that g ∼ f , then also the limit limT→∞

1T

∫ T0 g(t)dt

exists and equals h−1L.

As a direct consequence of the Renewal theorem and Lemma 3.1.2 we obtain

Corollary 3.1.3. Under the assumptions of Theorem 3.1.1 the following limit alwaysexists and is equal to the expression on the right hand side:

limT→∞

1

T

∫ T

0Z(t)dt =

1

η

∫ ∞

−∞z(τ)dτ.

Proof. If y1, ..., yN is h-arithmetic, just note that the function f(t) = hη

∑∞k=−∞ z(t −

kh) in Theorem 3.1.1(ii) is uniformly bounded and periodic, and apply Lemma 3.1.2(ii)to g(t) = Z(t). In the non-arithmetic case the limit limt→∞ Z(t) exists and the assertionfollows by applying Lemma 3.1.2 to g(t) = Z(t) which is asymptotic to the constantfunction f ≡ limt→∞ Z(t).

44

Reformulation of the Renewal theorem. Now we are ready to restate the Renewaltheorem in a more convenient way. Here we will always consider some self-similar setF with contraction ratios ri and similarity dimension s. Therefore we fix pi = rs

i andyi = − log ri. We substitute t = − log ǫ, since we are interested in the limiting behaviourof functions f : (0,∞) → R as the argument ǫ tends to zero. Moreover, by taking intoaccount Corollary 3.1.3, we conclude the existence of average limits from the asymptoticperiodicity.

Theorem 3.1.4. (Adapted Renewal theorem)Let F be a self-similar set with ratios r1, . . . , rN and similarity dimension s. For afunction f : (0,∞) → R, suppose that for some k ∈ R the function ϕk defined by

ϕk(ǫ) = f(ǫ) −N∑

i=1

rki 1(0,ri](ǫ)f(ǫ/ri) (3.1.6)

has a discrete set of discontinuities and satisfies

|ϕk(ǫ)| ≤ cǫk−s+γ (3.1.7)

for some constants c, γ > 0 and all ǫ > 0. Then ǫs−kf(ǫ) is uniformly bounded in (0,∞)and the following holds:

(i) The limit limδ→0

1|log δ|

∫ 1δ ǫs−kf(ǫ)dǫ

ǫ exists and equals

1

η

∫ 1

0ǫs−k−1ϕk(ǫ) dǫ , (3.1.8)

where η = −∑Ni=1 rs

i log ri.

(ii) If − log r1, . . . ,− log rN is non-arithmetic, then the limit of ǫs−kf(ǫ) as ǫ → 0exists and equals the average limit.

Proof. The definition (3.1.6) of ϕk implies that

f(ǫ) =N∑

i=1

rki 1(0,ri](ǫ)f(ǫ/ri) + ϕk(ǫ) . (3.1.9)

Define

Z(t) =

e(k−s)tf(e−t) for t ≥ 0

0 for t < 0. (3.1.10)

Taking into account (3.1.9), for all t ≥ 0 we have

Z(t) = e(k−s)t

(N∑

i=1

rki 1(0,ri](e

−t)f(e−(t+log ri)) + ϕk(e−t)

)

=N∑

i=1

rsi e

(k−s)(t+log ri)1[− log ri,∞)(t)f(e−(t+log ri)) + e(k−s)tϕk(e−t)

45

The i-th term of the sum can be replaced by rsi Z(t + log ri). (For t < − log ri this

expression equals zero as well as the corresponding i-th term in the sum.) Thus we havethe renewal equation

Z(t) =N∑

i=1

rsi Z(t + log ri) + z(t) , (3.1.11)

where the function z : R → R is defined by

z(t) =

e(k−s)tϕk(e

−t) for t ≥ 00 for t < 0

. (3.1.12)

Observing that the assumptions on ϕk ensure z to have a discrete set of discontinuitiesand to satisfy

|z(t)| = e(k−s)t∣∣ϕk(e

−t)∣∣ ≤ ce−tγ

for some constants c, γ > 0 and all t ≥ 0, we can apply the Theorem 3.1.1 with pi = rsi

and yi = − log ri. There are two cases to discuss.

The non-arithmetic case. If − log r1, . . . ,− log rN is non-arithmetic, then the limit

limt→∞

Z(t) = limt→∞

e−t(s−k)f(e−t) = limǫ→0

ǫs−kf(ǫ)

exists and is equal to the integral

1

η

∫ ∞

0z(τ)dτ . (3.1.13)

By (3.1.12) and with the substitution r = e−τ we obtain

limǫ→0

ǫs−kf(ǫ) =1

η

∫ 1

0rs−k−1ϕk(r)dr .

This completes the proof of (ii) of Theorem 3.1.4. For the non-arithmetic case, (i)immediately follows from (ii), since the average limit exists whenever the limit existsand both coincide.

The arithmetic case. If − log r1, . . . ,− log rN is h-arithmetic for some h > 0, Corol-lary 3.1.3 states that the limit

limT→∞

1

T

∫ T

0Z(t) dt = lim

δ→0

1

|log δ|

∫ 1

δǫs−kf(ǫ)

dǫ

ǫ

exists and equals the integral in (3.1.13). Therefore we obtain formula (3.1.8) also forthe h-arithmetic case. This completes the proof of Theorem 3.1.4.

Remark 3.1.5. In the h-arithmetic case, Theorem 3.1.4 concentrates on the existenceof average limits. However, some additional information on the limiting behaviour canbe derived from the original Renewal theorem, which stated the existence of some (ad-ditively) periodic function of period h, to which Z(t) is asymptotic as t → ∞. In thesituation of Theorem 3.1.4 this is translated into the existence of a (multiplicatively)periodic function G(ǫ) of period ζ = e−h, i.e. G(ζǫ) = G(ǫ) for all ǫ > 0, to which thefunction g(ǫ) = ǫs−kf(ǫ) is asymptotic as ǫ → 0.

46

4. Fractal curvatures – proofs

The purpose of this chapter is to provide proofs of the results presented in Section 1.2.After some preparations and the proof of Proposition 1.2.1 in the first section, we statesome key estimate (Lemma 4.2.1) which we will then use to prove Theorem 1.2.6. Theproblem left is the verification of the key estimate, which will be done in Sections 4.3 and4.4. In 4.5 we then turn to the proofs of Theorem 1.2.2 and Theorem 1.2.8. Finally, inSection 4.6 we reproof Gatzouras’s theorem on the existence of the (average) Minkowskicontent.

Throughout the chapter we assume F to be a self-similar set in Rd satisfying OSC.

Moreover, O will always denote some feasible open set of F such that the SOSC issatisfied, i.e. in particular F ∩ O 6= ∅ .

4.1. Preparations

We introduce some more notation required in the proofs below and collect some simplefacts. On the way we prove Proposition 1.2.1.

Code space and level sets. Set Σ := 1, . . . , N and for n = 0, 1, 2, . . . let Σn denotethe set of all sequences w1w2 . . . wn such that wi ∈ Σ . n is called the length of thesequence w = w1w2 . . . wn . We set

Σ∗ :=∞⋃

n=0

Σn.

Observe that Σ∗ contains all finite sequences including the empty word w ∈ Σ0 . Finitesequences w ∈ Σ∗ are also called words over the alphabet Σ. If v = v1 . . . vm andw = w1 . . . wn are words in Σ∗ , then vw simply denotes the word v1 . . . vmw1 . . . wn .Moreover, let Σ∞ denote the family of all infinite sequences w1w2w3 . . . such that wi ∈ Σ .For w = w1 . . . wn ∈ Σ∗ , we introduce the abbreviations

rw := rw1rw2

. . . rwn

and

Sw := Sw1 Sw2

. . . Swn .

The sets SwF are called the level sets of F . Since the Si are contractions and sinceSwiF ⊂ SwF for each w ∈ Σ∗ and i ∈ Σ , the mapping π : Σ∞ → F given by

w1w2w3 . . . 7→ x :=∞⋂

n=1

Sw1w2...wnF

47

is well defined. Observe that π is surjective but not necessarily injective, i.e. for eachx ∈ F there exists some sequence w ∈ Σ∞ such that x = π(w) but it might not beunique.

The families Σ(r). For 0 < r ≤ 1, let Σ(r) be the family of all finite words w =w1 . . . wn ∈ Σ∗ such that

rw < r ≤ rwr−1wn

. (4.1.1)

For convenience, we define Σ(r), for r > 1, to be the set containing only the emptyword. It is clear that for fixed r ≤ 1 for each sequence w1w2w3 . . . ∈ Σ∞ there is exactlyone n such that w = w1 . . . wn satisfies (4.1.1) (and, for r > 1, n = 0, correspondingly).Therefore, on the one hand

F =⋃

w∈Σ(r)

SwF (4.1.2)

for each r > 0. On the other hand the words in Σ(r) are mutually incompatible, i.e.there is no pair of words v, w ∈ Σ(r) such that v = ww′ for some nonempty w′ ∈ Σ∗.Moreover, for each r > 0, ∑

w∈Σ(r)

rsw = 1, (4.1.3)

which is easily seen from the definition of the similarity dimension s. Due to (4.1.1),Σ(r) consists of words w, for which the corresponding level sets SwF are approximatelyof the same size r · diamF . Note that (4.1.1) implies in particular

rw < r ≤ rwr−1min (4.1.4)

for each w ∈ Σ(r), where rmin = miniri.The cardinalities #Σ(r) of these finite families of words are bounded as follows. By

(4.1.4), rrmin ≤ rw < r for each w ∈ Σ(r). Hence, by (4.1.3), on the one hand

1 =∑

w∈Σ(r)

rsw <

∑

w∈Σ(r)

rs = rs#Σ(r)

and on the other hand

1 =∑

w∈Σ(r)

rsw ≥

∑

w∈Σ(r)

(rrmin)s = rs

minrs#Σ(r) .

Therefore,r−s < #Σ(r) ≤ r−s

minr−s. (4.1.5)

The families Σ(r) will play an important role in the proofs later on. As a first applicationof these families, we present a proof of Proposition 1.2.1.

Proof of Proposition 1.2.1. Assume Fǫ ∈ Rd . Then, by Fact 1.1.1 , Fδ ∈ Rd for allδ ≥ ǫ . Let now δ < ǫ and set r = ǫ−1δ . By (4.1.2), we have

Fδ =⋃

w∈Σ(r)

(SwF )δ =⋃

w∈Σ(r)

SwFδ/rw.

Since δ/rw > δ/r = ǫ , Fδ/rwis a parallel set of Fǫ and thus, again by Fact 1.1.1 ,

polyconvex. Hence each set SwFδ/rwin the finite union above is polyconvex implying

the same for Fδ .

48

Definition of u, ρ and γ. Above we fixed some feasible open set O of F such thatthe SOSC is satisfied. The condition F ∩ O 6= ∅ implies that there exists a sequenceu = u1 . . . up ∈ Σ∗ such that

SuF ⊂ O (4.1.6)

and, since SuF is compact, some constant α > 0 such that

d(x, ∂O) > α for all x ∈ SuF.

Applying the similarity Sw, w ∈ Σ∗, the above inequality yields

d(x, ∂SwO) > αrw for all x ∈ SwuF. (4.1.7)

Define

ρ := rminα

2. (4.1.8)

Moreover, for each ǫ > 0 we set ǫ∗ = ρ−1ǫ. As will become clear later, it is veryconvenient to look at the level sets with w ∈ Σ(ǫ∗) when investigating ǫ-parallel sets.

Finally we introduce the following numbers. Choose some r such that u ∈ Σ(r) andlet s be the unique solution of ∑

v∈Σ(r), v 6=u

rsv = 1 (4.1.9)

and γ := s−s. By (4.1.3), s < s and so γ > 0. Obviously, γ and s depend on the word uwe fixed above. Throughout Chapters 4 and 5 we consider the word u and the constantγ together with the set O as being once and for all fixed for the self-similar set F .

4.2. Proof of Theorem 1.2.6

Throughout this section we assume that the self-similar set F has polyconvex parallelsets, as demanded in the hypothesis of Theorem 1.2.6. Moreover, let k ∈ 0, 1, . . . , dbe fixed.

The key estimate. For each r > 0, we define the set

O(r) :=⋃

v∈Σ(r)

SvO, (4.2.1)

where O is the feasible open set for F we fixed above. Observe that O(r) is again afeasible open set of F for each r > 0. In particular, O = O(r) for any r > 1 andO(1) = SO =

⋃i SiO. For the complement O(r)c of these sets the following estimate

holds.

Lemma 4.2.1. For each r > 0, there exists a constant c > 0 such that for all ǫ ≤ δ ≤ ρr

Cvark (Fǫ, (O(r)c)δ) ≤ cǫk−sδγ .

49

This estimate roughly means that, as δ and ǫ approach 0, the k-th curvature con-centrates more and more in the set O(r)−δ, the inner parallel set of O(r), while thecurvature in the complement (O(r)−δ)

c = (O(r)c)δ vanishes. The constants ρ and γ arethose we fixed in (4.1.8) and (4.1.9). c depends on r (and the k fixed above) but is inde-pendent of ǫ and δ. Since C±

k (Fǫ, (O(r)c)δ) ≤ Cvark (Fǫ, (O(r)c)ǫ) and |Ck(Fǫ, (O(r)c)ǫ)| ≤

Cvark (Fǫ, (O(r)c)ǫ), the above Lemma provides also upper bounds for these expressions.

We require the key estimate in this full generality in the proofs on the weak convergenceof curvature measures in Chapter 5. For the moment the following special version issufficient, where we set ǫ = δ and also fix r = 1.

Corollary 4.2.2. There exist some constant c > 0 such that for all 0 < ǫ ≤ 1

Cvark (Fǫ, ((SO)c)ǫ) ≤ cǫk−s+γ .

Proof. Setting in Lemma 4.2.1 r = 1, i.e. O(r) = SO, and ǫ = δ, the validity of thestated inequality follows immediately for all ǫ ≤ ρ. If necessary, the constant c can beenlarged such that it also holds for ρ < ǫ ≤ 1.

Note that, for ǫ fixed, the estimate remains valid with (SO)c)ǫ replaced by any of itssubsets. In particular, since SO ⊆ O and thus (Oc)ǫ ⊆ ((SO)c)ǫ, the estimate holds aswell for the sets (Oc)ǫ.

We postpone the proof of Lemma 4.2.1 for the moment and first discuss how it can beused to prove Theorem 1.2.6. The first step is the investigation of the scaling functions.

Scaling functions. Recall from (1.2.1) that the k-th scaling function Rk is defined by

Rk(ǫ) = Ck(Fǫ) −N∑

i=1

1(0,ri](ǫ)Ck((SiF )ǫ),

for ǫ > 0. We investigate the properties of Rk to see that the Renewal theorem can beapplied. On the one hand we require an upper bound for the growth of |Rk| as ǫ → 0,which will be derived from Corollary 4.2.2, and on the other hand a statement on thecontinuity of Rk.

Lemma 4.2.3. There is a constant c > 0 such that for all 0 < ǫ ≤ 1

|Rk(ǫ)| ≤ cǫk−s+γ . (4.2.2)

Proof. For ǫ > 0, let U(ǫ) =⋃

i6=j(SiF )ǫ ∩ (SjF )ǫ and Bj(ǫ) = (SjF )ǫ \ U(ǫ). Then

Fǫ =⋃

j Bj(ǫ) ∪ U(ǫ) is a disjoint union and so

Ck(Fǫ) =N∑

j=1

Ck(Fǫ, Bj(ǫ)) + Ck(Fǫ, U(ǫ)).

Similarly,

Ck((SjF )ǫ) = Ck((SjF )ǫ, Bj(ǫ)) + Ck((SjF )ǫ, U(ǫ)),

50

since Bj(ǫ) ∩ (SiF )ǫ = ∅ for j 6= i. Thus the function Rk can be written as

Rk(ǫ) =N∑

j=1

(Ck(Fǫ, B

j(ǫ)) − Ck((SjF )ǫ, Bj(ǫ))

)+ Ck(Fǫ, U(ǫ)) −

N∑

j=1

Ck((SjF )ǫ, U(ǫ)).

Observe that the set Aj(ǫ) = (⋃

i6=j(SiF )ǫ)c is open and that Fǫ∩Aj(ǫ) = (SjF )ǫ∩Aj(ǫ).

Since Bj(ǫ) ⊆ Aj(ǫ), the locality property of Ck implies,

Ck(Fǫ, Bj(ǫ)) = Ck((SjF )ǫ, B

j(ǫ)).

Hence all terms of the first sum on the right hand side equal zero and can be omitted.Taking absolute values, we infer that

|Rk(ǫ)| ≤ |Ck(Fǫ, U(ǫ))| +N∑

j=1

|Ck((SjF )ǫ, U(ǫ))| . (4.2.3)

For the first term on the right hand side we claim that U(ǫ) ⊆ ((SO)c)ǫ and concludefrom Corollary 4.2.2, the existence of c > 0 such that for all ǫ > 0

|Ck(Fǫ, U(ǫ))| ≤ Cvark (Fǫ, U(ǫ)) ≤ cǫk−s+γ . (4.2.4)

For a proof of the set inclusion U(ǫ) ⊆ ((SO)c)ǫ, let x ∈ U(ǫ). We show that d(x, (SO)c) ≤ǫ and thus x ∈ ((SO)c)ǫ. Assume d(x, (SO)c) > ǫ. Since the union SO =

⋃i SiO

is disjoint, there is a unique j such that x ∈ SjO. Moreover, d(x, ∂SjO) > ǫ. Sincex ∈ U(ǫ), there is at least one index i 6= j such that x ∈ (SiF )ǫ and consequently a pointy ∈ SiF with d(x, y) ≤ ǫ. But then y ∈ SiF ∩ SjO, a contradiction to OSC. Hence,d(x, (SO)c) ≤ ǫ .

For the remaining terms in (4.2.3) observe that for each j

|Ck((SjF )ǫ, U(ǫ))| = rkj

∣∣∣Ck(Fǫ/rj, S−1

j U(ǫ))∣∣∣ ≤ rk

j Cvark (Fǫ/rj

, S−1j U(ǫ)).

We show that S−1j U(ǫ)∩Fǫ/rj

⊆ (Oc)ǫ/rj. Let x ∈ S−1

j U(ǫ)∩Fǫ/rj. Then Sjx ∈ U(ǫ) and

so there exists at least one index i 6= j with Sjx ∈ (SiF )ǫ. Hence d(Sjx, ∂SjO) ≤ ǫ sinceotherwise there would exist a point y ∈ SiF ∩ SjO, a contradiction to OSC. Therefore,d(x, ∂O) ≤ ǫ/rj , i.e. x ∈ (Oc)ǫ/rj

.By the set inclusion just proved and Corollary 4.2.2, there exists a constant c > 0 such

that for all ǫ, Cvark (Fǫ/rj

, S−1j U(ǫ)) is bounded from above by c(ǫ/rj)

k−s+γ and thus

|Ck((SjF )ǫ, U(ǫ))| ≤ cjǫk−s+γ

where cj := crs−γj .

Since each of the terms in (4.2.3) is bounded from above by cǫk−s+γ for some constantc > 0, we can also find such a constant for |Rk(ǫ)| and so the assertion follows.

The last missing ingredient for the application of Theorem 3.1.4 is a statement on thecontinuity properties of Rk. We require that Rk is continuous except for a discrete set,i.e. the discontinuities are well separated from each other and do not accumulate insidethe domain (0,∞) of Rk.

51

Lemma 4.2.4. The function Rk has a discrete set of discontinuities in (0,∞).

Proof. From Lemma 2.1.3 it is easily seen that Ck(Fǫ) and Ck((SiF )ǫ) have this property,since they have at most finitely many discontinuities in each interval [ǫ0,∞), ǫ0 > 0. ThusRk has at most finitely many discontinuities in each interval [ǫ0,∞) and the assertionfollows.

Note that the discontinuities of Rk possibly accumulate at 0.

Proof of Theorem 1.2.6. Since

Ck((SiF )ǫ) = rki Ck(Fǫ/ri

),

the functions f(ǫ) := Ck(Fǫ) and ϕk(ǫ) := Rk(ǫ) satisfy a renewal equation

ϕk(ǫ) = f(ǫ) −N∑

i=1

rki 1(0,ri](ǫ)f(ǫ/ri)

as in (3.1.6). Now Lemma 4.2.3 and Lemma 4.2.4 ensure that the hypotheses of Theo-rem 3.1.4 are satisfied. Therefore the average limit of the expression ǫs−kCk(Fǫ) existsand in case of a non-arithmetic set F also the limit.

To complete the proof of Theorem 1.2.6, it remains to verify the key estimate Lem-ma 4.2.1. This is the agenda for the succeeding two sections.

4.3. Convex representations of Fǫ

In this section we translate the problem of proving the key estimate into the problemof estimating cardinalities of certain families of level sets. For getting control over thebehaviour of the measure Cvar

k (Fǫ, · ) as ǫ → 0, we decompose Fǫ into convex sets foreach ǫ > 0. The idea is to use small copies from a fixed collection of convex sets Ki forthe decomposition, such that only the number of convex sets used in the decompositionincreases as ǫ → 0 while their curvatures are ”fixed” (up to scaling). This allows toreduce the problem to estimating the number of sets involved in such representations.

First we fix the collection of convex sets that will be used. It is convenient to use arepresentation of Fρ by convex sets, where ρ is the constant we defined in (4.1.8).

Decomposition of parallel sets. Let Ki, i = 1, . . . , P be convex sets such that

Fρ =P⋃

i=1

Ki.

Note that for each ǫ > ρ this provides a decomposition of Fǫ into convex sets, namelyinto parallel sets of the Ki. If ǫ = ρ + δ then

Fǫ =P⋃

i=1

Kiδ. (4.3.1)

52

For ǫ < ρ the decomposition is done in two steps. First we decompose Fǫ into smallcopies of Fρ :

Fǫ =⋃

w∈Σ(ǫ∗)

(SwF )ǫ .

The choice of w from the family Σ(ǫ∗) ensures that ρ < ǫrw

≤ ρr−1min (cf. (4.1.4)) and so

(SwF )ǫ = SwFǫ/rw= Sw(Fρ)δ for some 0 ≤ δ < δmax where δmax := ρ(r−1

min − 1). Hence(SwF )ǫ has a representation by small copies of δ-parallel sets of Ki. For each w ∈ Σ(ǫ∗),

(SwF )ǫ =P⋃

i=1

SwKiδ for some 0 ≤ δ < δmax. (4.3.2)

This also provides a representation of Fǫ as a union of convex sets.

Intersection numbers. Now the first task is to investigate finite intersections of de-composition sets (SwF )ǫ. While for the existence of a representation (4.3.2) it was onlyimportant that ǫ is not too small compared to rw (rw < ǫ∗), the reversed relation that ǫis also not too large compared to rw (ǫ∗ ≤ rwr−1

min) will now be essential for the control ofintersection numbers. Recall the definition of the intersection number of a finite familyof sets from (2.2.4).

The first statement says, that the intersection number of the family of sets (SwF )ǫ

with w ∈ Σ(ǫ∗) is uniformly bounded (independent of ǫ).

Lemma 4.3.1. There exists a constant Γmax such that for each ǫ > 0 and r ≥ ǫ∗

Γ((SwF )ǫ : w ∈ Σ(r)) ≤ Γmax.

Proof. Note that it suffices to prove the assertion for r = ǫ∗, since choosing for fixedr > 0 some ǫ < ρr (i.e. r > ǫ∗) does not increase the intersection number compared tothe choice ǫ = ρr. Fix ǫ > 0 and recall the definition of the word u from (4.1.6). Firstwe show that

(i) The sets (SwuF )ǫ, w ∈ Σ(ǫ∗), are pairwise disjoint.By (4.1.7), we have

d(x, ∂SwO) > αrw ≥ αrminǫ∗ ≥ ǫ

for each w ∈ Σ(ǫ∗) and x ∈ SwuF . This implies

(SwuF )ǫ ⊆ SwO.

Since, by OSC, the sets SwO, w ∈ Σ(ǫ∗), are pairwise disjoint, assertion (i) follows.Fix some v ∈ Σ(ǫ∗) and a point x ∈ SvuF . Let Γ(v) denote the number of sequences

w ∈ Σ(ǫ∗) with (SwF )ǫ ∩ (SvF )ǫ 6= ∅. Then it holds(ii) For each sequence w counted in Γ(v), the set (SwuF )ǫ is contained in the ball

B(x, cǫ) where c := 2ρ−1diamF + 3. Note that c is independent of v or ǫ.Let y ∈ (SwuF )ǫ. Since (SwF )ǫ ∩ (SvF )ǫ 6= ∅, there is a point x′ in this intersection.Therefore

d(x′, y) ≤ diam (SwF )ǫ = rwdiamF + 2ǫ ≤ ǫ∗diamF + 2ǫ = (ρ−1diamF + 2)ǫ,

53

since w ∈ Σ(ǫ∗), and similarly

d(x, x′) ≤ diam (SvF ) + ǫ ≤ (ρ−1diamF + 1)ǫ.

Thus d(x, y) ≤ (2ρ−1diam F + 3)ǫ = cǫ for all y ∈ (SwuF )ǫ, and so (SwuF )ǫ ⊆ B(x, cǫ)as stated in (ii).

Observing that each ǫ-parallel set contains an ǫ-ball and has thus Lebesgue measureat least κdǫ

d, where κj denotes the volume of the j-dimensional unit ball in Rj , and

taking into account (i) and (ii), we obtain

Γ(v)κdǫd ≤ λd(B(x, cǫ)) = κd(cǫ)

d.

Hence Γ(v) ≤ cd =: Γmax, where Γmax is independent of ǫ, as desired.

Curvature of intersections of level sets. Lemma 4.3.1 allows to bound the variationmeasures of arbitrary intersections of sets (SwF )ǫ with w ∈ Σ(ǫ∗).

Lemma 4.3.2. There is a constant c > 0 such that for all ǫ > 0 and all Borel setsB ⊆ R

d

Cvark ((Sw(1)F )ǫ ∩ . . . ∩ (Sw(m)F )ǫ, B) ≤ cǫk

whenever m ∈ N and w(1), . . . , w(m) ∈ Σ(ǫ∗) .

Proof. It suffices to show that the total masses Cvark ((Sw(1)F )ǫ ∩ . . . ∩ (Sw(m)F )ǫ) sat-

isfy the inequality for some c > 0. Moreover, we can assume m ≤ Γmax, since, byLemma 4.3.1, (Sw(1)F )ǫ ∩ . . . ∩ (Sw(m)F )ǫ = ∅ for m > Γmax.

Applying Lemma 2.2.5 to the sets Xj := (Sw(j)F )ǫ, which have representations (4.3.2)by P convex sets Kj,i := Sw(j)K

iδ(j) for some δ(j) with 0 ≤ δ(j) < δmax, we obtain for

the set X :=⋃m

j=1 Xj

Cvark (X) ≤ (2(P m) − 1)max

j,iCk(K

j,i). (4.3.3)

Observe now that Kiδ(j) ⊂ Ki

δmaxand so the monotonicity of Ck for convex sets and the

scaling property imply

Ck(Kj,i) = rk

w(j)Ck(Kiδ(j)) ≤ rk

w(j)Ck(Kiδmax

).

Since w(j) ∈ Σ(ǫ∗), rw(j) ≤ ǫ∗ and so

Ck(Kj,i) ≤ ρ−kǫkCk(K

iδmax

).

Since the right hand side does not depend on j, the maximum in (4.3.3) is boundedfrom above by ρ−kǫk maxi Ck(K

iδmax

). Noting that m ≤ Γmax it follows that the asserted

inequality is satisfied for the constant c := (2(PΓmax)−1)ρ−k maxi Ck(Kiδmax

), which doesneither depend on ǫ nor on m or the choice of the sequences w(j). This completes theproof.

54

Curvature estimates via cardinalities. Using the above estimate for finite intersectionsof level sets and the intersection number Γmax we can now reduce the task of estimatingCvar

k (Fǫ, · ) to the problem of determining the cardinalities of certain families of levelsets. For a closed set B ⊆ R

d and ǫ > 0, let

Σ(B, ǫ) = w ∈ Σ(ǫ∗) : (SwF )ǫ ∩ B 6= ∅ . (4.3.4)

Lemma 4.3.3. There is a constant c′ > 0 such that for all closed sets B ⊆ Rd and all

ǫ > 0

Cvark (Fǫ, B) ≤ c′ #Σ(B, ǫ)ǫk.

Proof. Fix ǫ > 0. Observe that each (SwF )ǫ with w ∈ Σ(ǫ∗) which does not intersectB has some positive distance to B. Hence there is an open set A containing B suchthat Fǫ ∩ A =

⋃w(SwF )ǫ ∩ A where the union is taken over all w ∈ Σ(B, ǫ). Hence the

locality of the curvature measure implies

Cvark (Fǫ, B) = Cvar

k (⋃

w∈Σ(B,ǫ)

(SwF )ǫ, B).

The union on the right hand side consists of m = #Σ(B, ǫ) polyconvex sets. It satisfiesthe conditions of Corollary 2.2.4, since, by the Lemma 4.3.2, there exist upper boundsb := cǫk (ǫ is fixed) for the variation measures with respect to finite intersections. More-over, by Lemma 4.3.1, Γmax is an upper bound for the intersection number of the family(SwF )ǫ : w ∈ Σ(B, ǫ). Thus, by Corollary 2.2.4, the assertion holds for the constantc′ = 2Γmaxc, where c is the constant of Lemma 4.3.2.

4.4. Cardinalities of level set families

In view of Lemma 4.3.3 it is evident that in order to prove Lemma 4.2.1 we requireupper bounds for the cardinalities of the families Σ(B, ǫ) for the sets B = (O(r)c)δ.Such bounds will be discussed now. Note that in this section no curvature is involved.Therefore, here the assumption of polyconvex parallel sets for F is not required. Themain result of this section is the following:

Lemma 4.4.1. For each r > 0, there is a constant c > 0 such that for all 0 < ǫ ≤ δ ≤ ρr

#Σ((O(r)c)δ, ǫ) ≤ cǫ−sδγ .

Again the constant γ is as defined in (4.1.9). Note that the above estimate remainsvalid with the set (O(r)c)δ replaced by any of its subsets. Lemma 4.2.1 follows immedi-ately.

Proof of Lemma 4.2.1. Combine Lemma 4.3.3 and Lemma 4.4.1 and note that theconstant c′ in Lemma 4.3.3 is independent of the choice of the set B.

55

Splitting the proof of Lemma 4.4.1. It remains to provide a proof of Lemma 4.4.1,which we will divide into several steps. For this purpose we introduce some more nota-tion. We say that a word v ∈ Σ∗ occurs in a word w ∈ Σ∗, in symbols v ⊂ w, if thereare words w′, w′′ ∈ Σ∗ such that w = w′vw′′. We write v /⊂ w, if v does not occur in w.

Recall the definition of the word u = u1 . . . up we defined in (4.1.6). The main ideaof the proof is that some level set (SwF )ǫ for which u occurs in w lies sufficiently faraway from the boundary of O(r) and is thus not counted in the family #Σ ((O(r)c)δ, ǫ)if ǫ, δ, r are arranged appropriately. The problem then reduces to counting the numberof words w in certain families such that u does not occur in w.

For ǫ > 0 let Ξ(ǫ) be the family of all words w ∈ Σ(ǫ) such that u does not occur inw, i.e.

Ξ(ǫ) = w ∈ Σ(ǫ) : u /⊂ w .

For 0 < ǫ ≤ δ and w ∈ Σ(ǫ) there exists a subword w′ ∈ Σ(δ) such that w = w′w′′ forsome w′′ ∈ Σ∗. So, for 0 < ǫ ≤ δ, let

Ω(ǫ, δ) =w ∈ Σ(ǫ) : w = w′w′′, w′ ∈ Σ(δ), u /⊂ w′ . (4.4.1)

Similarly, for 0 < ǫ ≤ δ ≤ r, let

Λ(ǫ, δ, r) =w ∈ Σ(ǫ) : w = w0w′w′′, w0 ∈ Σ(r), w0w′ ∈ Σ(δ), u /⊂ w′ . (4.4.2)

Then the following relations hold for the cardinalities of these families. Recall thatǫ∗ = ρ−1ǫ.

I. For all ǫ∗ ≤ δ∗ ≤ r, #Σ ((O(r)c)δ, ǫ) ≤ #Λ(ǫ∗, δ∗, r).

II. For all ǫ ≤ δ ≤ r, #Λ(ǫ, δ, r) ≤ #Σ(r)#Ω( ǫr , δ

rrmin).

III. For all ǫ ≤ δ, #Ω(ǫ, δ) ≤ r−smin

(ǫδ

)−s#Ξ(δ).

IV. There exist c1, γ > 0 such that, #Ξ(ǫ) ≤ c1ǫγ−s.

Combining these estimates, Lemma 4.4.1 is easily derived. Fix r > 0. Combining theinequalities II. – IV., we derive for ǫ ≤ δ ≤ r

#Λ(ǫ, δ, r) ≤ #Σ(r)#Ω(ǫ

r,

δ

rrmin)

≤ #Σ(r)r−smin

(ǫrmin

δ

)−s#Ξ(

δ

rrmin)

≤ #Σ(r)r−2smin ǫ−sδsc1(rrmin)

s−γδγ−s

≤ c2ǫ−sδγ ,

where the constant c2 only depends on r (c2 = c1r−s−γmin #Σ(r)rs−γ). Applying this to

the right hand side of I, we obtain for ǫ∗ ≤ δ∗ ≤ r

#Σ((O(r)c)δ, ǫ) ≤ #Λ(ǫ∗, δ∗, r) ≤ c2(ρ−1ǫ)−s(ρ−1δ)γ = cǫ−sδγ

where c = c2ρs−γ is independent of δ and ǫ. Hence we have derived a constant c satisfying

the assertion of Lemma 4.4.1.It remains to provide proofs of the four inequalities I. – IV.

56

Proof of I. Let 0 < ǫ∗ ≤ δ∗ ≤ r and w ∈ Σ(ǫ∗) \ Λ(ǫ∗, δ∗, r), i.e w ∈ Σ(ǫ∗) andw = w0w′w′′ such that w0 ∈ Σ(r) and w0w′ ∈ Σ(δ∗) but u ⊂ w′. We show that thisimplies (SwF )ǫ ∩ (O(r)c)δ = ∅ and thus w /∈ Σ((O(r)c)δ, ǫ), proving the assertion.

Let x ∈ (SwF )ǫ. There exists y ∈ SwF with d(x, y) ≤ ǫ. The assumption u ⊂ w′

implies that there exist u′, u′′ ∈ Σ∗ such that w = u′uu′′. By definition of u, the setinclusions

SwF ⊆ Sw0u′uF ⊂ Sw0u′O ⊆ Sw0O ⊆ O(r)

hold and so y ∈ SwF is an interior point of the open set O(r). We estimate its distanceto the boundary and thus to the complement of O(r). Taking into account (4.1.7) andw0w′ ∈ Σ(δ∗), we infer

d(y, ∂O(r)) ≥ d(y, ∂Sw0u′O) > rw0u′α

≥ 2ρr−1minrw0w′ ≥ 2ρδ∗ = 2δ.

Hence d(x, O(r)c) ≥ d(y, O(r)c) − d(x, y) > 2δ − ǫ ≥ δ, implying x /∈ (O(r)c)δ.

Proof of II. From the definitions it is easily seen that

#Λ(ǫ, δ, r) =∑

w0∈Σ(r)

#Ω(ǫ

rw0

,δ

rw0

).

Now observe that for fixed δ, #Ω(ǫ, δ) is a decreasing function of ǫ (provided 0 < ǫ ≤ δ),and for fixed ǫ, #Ω(ǫ, δ) is increasing in δ (as long as ǫ ≤ δ). Therefore, in #Ω( ǫ

rw0

, δrw0

),ǫ

rw0

can be replaced by the smaller value ǫr , and independently δ

rw0

by the larger valueδ

rminr to provide the upper bound #Ω( ǫr , δ

rrmin), for each w0 ∈ Σ(r). (w0 ∈ Σ(r) implies

rw0 < r ≤ rw0r−1min and so 1

r < 1rw0

≤ 1rrmin

.) Since the obtained upper bound is

independent of w0 ∈ Σ(r), we conclude that #Λ(ǫ, δ, r) ≤ #Σ(r)#Ω( ǫr , δ

rrmin) as asserted

above.

Proof of III. Let ǫ ≤ δ. If w = w′w′′ ∈ Ω(ǫ, δ), then necessarily w′ ∈ Ξ(δ) andw′′ ∈ Σ( ǫ

rw′

). Therefore,

#Ω(ǫ, δ) =∑

w′∈Ξ(δ)

#Σ(ǫ

rw′

).

Observing now that #Σ(ǫ is a nonincreasing function of ǫ, we can replace ǫrw′

by the

smaller value ǫδ (w′ ∈ Ξ(δ) implies rw′ < δ and so ǫ

δ < ǫrw′

) to see that each term in the

above sum is bounded from above by #Σ( ǫδ ), which is independent of w′ and can thus

be taken out of the sum. Hence

#Ω(ǫ, δ) ≤ #Ξ(δ)#Σ(ǫ

δ).

Now we infer from (4.1.5), that #Σ( ǫδ ) ≤ r−s

min

(ǫδ

)−s, proving assertion III.

57

Proof of IV. We have to show that the expression

ξ(ǫ) := ǫs#Ξ(ǫ)

is bounded as ǫ → 0, where s = s−γ as in (4.1.9). Observe that #Ξ(ǫ) is a nonincreasing,nonnegative function of ǫ. Fix some r > 0, such that u ∈ Σ(r). For ǫ ≤ r, each wordw ∈ Ξ(ǫ) begins with some word v ∈ Σ(r) different from u and so

#Ξ(ǫ) ≤∑

v∈Σ(r),v 6=u

#Ξ(ǫ

rv) (4.4.3)

for all ǫ ≤ r.By (4.4.3), ξ(ǫ) satisfies for all ǫ ≤ r

ξ(ǫ) ≤∑

v∈Σ(r),v 6=u

rsvξ(

ǫ

rv),

and so, for all ǫ′ ≤ r,

supǫ≥ǫ′

ξ(ǫ) ≤ supǫ≥ǫ′

∑

v∈Σ(r),v 6=u

rsvξ(

ǫ

rv)

≤∑

v∈Σ(r),v 6=u

rsv sup

ǫ≥ǫ′ξ(

ǫ

rv)

≤ supǫ≥r−1ǫ′

ξ(ǫ).

Since #Ξ(ǫ) and thus ξ(ǫ) are bounded on each interval [ǫ′,∞), by the above inequality,ξ(ǫ) is bounded as ǫ → 0. This completes the proof of IV.

Remark 4.4.2. Families of finite sequences similar to Ξ(ǫ) have been studied by StevenP. Lalley in [18]. In the above proofs, especially in the proof of IV, we adopted some ofhis ideas.

4.5. Bounds for the total variations

In this section we prove the Theorems 1.2.2 and 1.2.8, which provide upper and lowerestimates, respectively, for the total mass Cvar

k (Fǫ) of the total variation measure of Fǫ.While the proof of Theorem 1.2.2 is based on the key estimate Lemma 4.2.1, for theone of Theorem 1.2.8 we only require appropriate decompositions of the parallel sets Fǫ.Throughout the section we assume that Fǫ ∈ Rd.

More scaling functions. In analogy with the k-th scaling function Rk we define func-tions R+

k , R−k and Rvar

k by

R•k(ǫ) = C•

k(Fǫ) −N∑

i=1

1(0,ri](ǫ)C•k((SiF )ǫ), (4.5.1)

for • ∈ +,−, var and each ǫ > 0. With similar arguments as in the proof ofLemma 4.2.3, which provided an upper bound for |Rk(ǫ)|, we can show the followingestimate to hold for |Rvar

k (ǫ)|.

58

Lemma 4.5.1. There are constants c, γ > 0 such that for all ǫ ∈ (0, 1]

|Rvark (ǫ)| ≤ cǫk−s+γ .

Corresponding estimates hold for R+k and R−

k . Based on this lemma, we will nowprove the boundedness of the expression ǫs−kCvar

k (Fǫ).

Proof of Theorem 1.2.2. Since, by the scaling property, Cvark ((SiF )ǫ) = rk

i Cvark (Fǫ/ri

),

by multiplying ǫs−k, we derive from (4.5.1) that

ǫs−kCvark (Fǫ) =

N∑

i=1

rsi 1(0,ri](ǫ)

(ǫ

ri

)s−k

Cvark (Fǫ/ri

) + ǫs−kRvark (ǫ).

Define the function gk : (0,∞) → R by setting gk(ǫ) = ǫs−kCvark (Fǫ) for ǫ ∈ (0, 1] and

gk(ǫ) = 0 for ǫ > 1. Then, for each ǫ ∈ (0, 1], the above equation can be rewritten as

gk(ǫ) =

N∑

i=1

rsi gk(

ǫ

ri) + ǫs−kRvar

k (ǫ),

and so, by Lemma 4.5.1, the following inequality holds for some c > 0

gk(ǫ) ≤N∑

i=1

rsi gk(

ǫ

ri) + cǫγ .

Note that this inequality is also trivially satisfied for all ǫ > 1. Hence for each ǫ0 > 0,

supǫ∈(ǫ0,ǫ0r−1

max]

gk(ǫ) ≤N∑

i=1

rsi sup

ǫ∈(ǫ0,ǫ0r−1max]

gk(ǫ

ri) + sup

ǫ∈(ǫ0,ǫ0r−1max]

cǫγ

≤ supǫ∈(ǫ0r−1

max,∞]

gk(ǫ) + c(ǫ0r−1max)

γ

and sosup

ǫ∈(ǫ0,∞)gk(ǫ) ≤ sup

ǫ∈(ǫ0r−1max,∞)

gk(ǫ) + cr−γmaxǫ

γ0 .

Iterating this inequality, i.e. applying it repeatedly to the first term on the right handside, after n steps we arrive at

supǫ∈(ǫ0,∞)

gk(ǫ) ≤ supǫ∈(ǫ0r−n

max,∞)

gk(ǫ) + cn∑

j=1

(rγmax)

−jǫγ0 .

Now set ǫ0 = rnmax. Then the first term on the right hand side supǫ∈(1,∞) gk(ǫ) equals

zero and so for each n ∈ N:

supǫ∈(rn

max,∞)gk(ǫ) ≤ c

n∑

j=1

(rγmax)

n−j = cn∑

j=1

(rγmax)

j .

Letting n → ∞, the right hand side converges to M := c1−rγ

maxand so supǫ∈(0,∞) gk(ǫ) ≤

M . Hence the expression ǫs−kCvark (Fǫ) is uniformly bounded by M in the interval (0, 1],

as we asserted in Theorem 1.2.2.

59

Remark 4.5.2. As recorded in Conjecture 2.2.2, we are convinced that, for K ∈ Rd

and • ∈ +.−, var, C•k(Kǫ) has at most finitely many discontinuities in (0,∞). If this

conjecture was true then one could easily prove an analogue of Lemma 4.2.4 for thefunctions R•

k, i.e. one could show that these functions have a discrete set of disconti-nuities. This statement combined with Lemma 4.5.1 would allow to apply the Renewaltheorem directly to the functions f•(ǫ) := C•

k(Fǫ) and ϕ•k(ǫ) := R•

k(ǫ). We could derivea statement similar to Theorem 1.2.6 on the existence of rescaled (average) limits ofthe total masses C•

k(Fǫ) of the variation measures. This would allow to strengthen aswell the results on fractal curvature measures. The rescaled variation measures wouldconverge weakly in a similar way as stated in Theorem 1.4.1 for the rescaled curvaturemeasures themselves.

Proof of Theorem 1.2.8. Fix some k ∈ 0, . . . , d. Let B be a set as in the hypothesesof Theorem 1.2.8, i.e. assume that there are constants ǫ0, β > 0 such that B ⊆ O−ǫ0 andCvar

k (Fǫ, B) ≥ β for ǫ ∈ (rminǫ0, ǫ0].Since for each r > 0 the sets SwO, w ∈ Σ(r), are pairwise disjoint, the same holds for

their subsets SwB and so for arbitrary ǫ > 0

Cvark (Fǫ) ≥

∑

w∈Σ(r)

Cvark (Fǫ, SwB).

Fix some ǫ < ǫ0 and choose r = r−1minǫ

−10 ǫ. Then, for each w ∈ Σ(r), rw < r−1

minǫ−10 ǫ ≤

rwr−1min, i.e. in particular ǫ ≤ rwǫ0, and so, since B ⊆ O−ǫ0 ,

SwB ⊆ SwO−ǫ0 = (SwO)−rwǫ0 ⊆ (SwO)−ǫ.

Hence, by the locality property of Cvark in the open set (SwO)−ǫ (where Fǫ ∩ (SwO)−ǫ =

(SwF )ǫ ∩ (SwO)−ǫ) and the scaling property,

Cvark (Fǫ, SwB) = Cvar

k (Sw(Fǫr−1w

), SwB) = rkwCvar

k (Fǫr−1w

, B).

Since ǫr−1w ∈ (rminǫ0, ǫ0], the hypothesis implies that Cvar

k (Fǫr−1w

, B) ≥ β and therefore,

Cvark (Fǫ) ≥

∑

w∈Σ(r)

rkwCvar

k (Fǫr−1w

, B) ≥∑

w∈Σ(r)

(ǫ−10 ǫ)kβ = βǫ−k

0 ǫk#Σ(r).

Recalling from (4.1.5) that #Σ(r) ≥ r−s = rsminǫ

s0ǫ

−s, we obtain

Cvark (Fǫ) ≥ βrs

minǫs−k0 ǫ−s+k = cǫ−s+k.

Since ǫ < ǫ0 was arbitrary, the assertion of Theorem 1.2.8 immediately follows.

4.6. Proof of Gatzouras’s theorem

Towards the end of Section 1.2 we discussed Gatzouras’s results on Minkowski measura-bility, which we want to prove now. In the proof we will use the results we have alreadyobtained, in particular Theorems 1.2.6 and 1.2.8. Going again through the proofs of

60

these theorems for k = d, we will check that, when replacing Cd(Fǫ, ·) with λd(Fǫ ∩ · ),most arguments remain valid even if the assumption Fǫ ∈ Rd is dropped. Only fewarguments have to be modified.

Let F be a self-similar set satisfying OSC. We emphasize that now we do not assumethe parallel sets Fǫ to be polyconvex. The main step towards a proof of Gatzouras’stheorem is a generalization of Lemma 4.3.3. Recall the definition of the family Σ(B, ǫ)from (4.3.4).

Lemma 4.6.1. There is a constant c > 0 such that for each closed sets B ⊆ Rd and all

ǫ > 0,λd(Fǫ ∩ B) ≤ c #Σ(B, ǫ) ǫd.

Proof. For ǫ > 0, the set inclusion Fǫ ∩ B ⊆ ⋃w∈Σ(B,ǫ)(SwF )ǫ implies that

λd(Fǫ ∩ B) ≤ λd(⋃

w∈Σ(B,ǫ)

(SwF )ǫ)

≤∑

w∈Σ(B,ǫ)

λd((SwF )ǫ)

≤∑

w∈Σ(B,ǫ)

rdwλd(Fǫ/rw

)

≤∑

w∈Σ(B,ǫ)

(ρ−1ǫ)dλd(Fρr−1

min

)

= #Σ(B, ǫ)ρ−dλd(Fρr−1

min

)ǫd,

where the third inequality is due to the scaling property and the last one to the fact thatrw < ρ−1ǫ ≤ rwr−1

min for each w ∈ Σ(B, ǫ). Therefore the constant c := ρ−dλd(Fρr−1

min

)

satisfies the assertion.

Since in Lemma 4.4.1 (as in the whole Section 4.4) we did not assume F to havepolyconvex parallel sets, we immediately obtain a generalization of the key estimateLemma 4.2.1, by combining Lemma 4.4.1 and the just derived Lemma 4.6.1.

Lemma 4.6.2. For each r > 0, there exists c > 0 such that for all ǫ ≤ δ ≤ ρr

λd(Fǫ ∩ (O(r)c)δ) ≤ cǫd−sδγ .

Note that this estimate will also be the key to the proof of Theorem 1.4.4, the localizedversion of Gatzouras’s theorem. Here, by setting r = 1 and δ = ǫ, we immediately derivean analogue of Corollary 4.2.2.

Corollary 4.6.3. There exist some constant c > 0 such that for all 0 < ǫ ≤ 1

λd(Fǫ, ((SO)c)ǫ) ≤ cǫd−s+γ .

It follows at once that Lemma 4.2.3 generalizes in a similar way. Just note thatfor the Lebesgue measure the locality property trivially holds for arbitrary Borel setsK, L, A ⊆ R

d, i.e. if A ∩ K = A ∩ L, then λd(K ∩ B) = λd(L ∩ B) for all Borel sets

61

B ⊆ A. Recall from (1.2.5) that in the general case the scaling function Rd was givenby

Rd(ǫ) = λd(Fǫ) −N∑

i=1

1(0,ri](ǫ)λd((SiF )ǫ) .

Lemma 4.6.4. There exists c > 0 such that for all 0 < ǫ ≤ 1

|Rd(ǫ)| ≤ cǫd−s+γ .

Since λd(Fǫ) is continuous in ǫ for each closed set F , the function Rd(ǫ) has at mostfinitely many discontinuities (at the points ri). Therefore the Renewal theorem 3.1.4applies. Taking into account that sd = s − d, it follows that M(F ) = Xd and fornon-arithmetic sets F also M(F ) = Xd.

It remains to show Xd > 0. For this observe that for k = d Theorem 1.2.8 generalizesas follows when the assumption Fǫ ∈ Rd is dropped.

Proposition 4.6.5. Let F be a self-similar set satisfying OSC and O some feasible openset of F . Suppose there exist some constants ǫ0, β > 0 and some Borel set B ⊂ O−ǫ0

such thatλd(Fǫ ∩ B) ≥ β

for each ǫ ∈ (rminǫ0, ǫ0]. Then for all ǫ < ǫ0

ǫs−dλd(Fǫ) ≥ c,

where c := βǫs−d0 rs

min > 0 .

Proof. Observe that the arguments in the proof of Theorem 1.2.8 remain valid in thegeneral case with Cvar

d (Fǫ, · ) = Cd(Fǫ, · ) replaced by λd(Fǫ ∩ · ).

It should be noted that the bound ǫs−dλd(Fǫ) ≥ c > 0 in Proposition 4.6.5 immediatelyimplies Xd > 0. Therefore it suffices to show that there is always some set B satisfyingthe hypothesis of this proposition. Let O be some feasible open set of F such thatthe SOSC holds. Then there exists a point x in F ∩ O 6= ∅ and, since O is open,some constant α′ > 0 such that d(x, ∂O) > α′. Let ǫ0 = α′

2 and B := B(x, ǫ0). ThenB ⊂ O−ǫ0 , since for each y ∈ B(x, ǫ0),

d(y, ∂O) ≥ d(x, ∂O) − d(x, y) > α′ − α′

2= ǫ0 .

Moreover, for each ǫ ∈ (rminǫ0, ǫ0],

λd(Fǫ ∩ B) ≥ λd(B(x, rminǫ0)) = κd(rminǫ0)d =: β .

Hence the hypothesis of Proposition 4.6.5 is satisfied and Xd ≥ c > 0 follows. Thiscompletes the proof of Gatzouras’s theorem.

62

5. Fractal curvature measures – proofs

In this chapter we will provide proofs of the results of Section 1.4, where we discussedweak limits of rescaled curvature measures. In the first section we will construct a familyof sets which on the one hand is adapted to the structure of F and on the other handis a separating class. This set family is the key to the proofs of all weak convergenceresults discussed here, since it allows to determine the limit measures uniquely, by com-puting their values for the sets of the family. The most part of Section 5.2 is reservedfor the proof of Theorem 1.4.1. In the end we will briefly outline how the argumentsin the proof of this theorem have to be adapted to obtain a proof of Theorem 1.4.2.In the last section we turn our attention to normalized curvature measures and proveTheorems 1.4.3 and 1.4.4. As before, F is some self-similar set satisfying OSC. The setO, the word u and the constants ρ and γ are as defined in Section 4.1.

5.1. A separating class for F

Let Bd denote the Borel σ-algebra of R

d. A family A of Borel sets is called a separatingclass if two measures that agree on A necessarily agree on B

d. We now introduce someseparating class AF , which is adapted to the structure of F . Define the set family

CF :=

C ∈ Bd : ∃r > 0 such that C ⊆ O(r)c

,

where O(r) is as defined in (4.2.1), and let

AF := SwO : w ∈ Σ∗ ∪ CF .

For the family AF the following holds.

Lemma 5.1.1. AF is an intersection stable generator of Bd.

Proof. The stability of the family AF with respect to intersections is easily seen. Eitherthe intersection of two sets SvO and SwO, v, w ∈ Σ∗, is empty or one of the sets iscontained in the other. Moreover, any intersection A ∩ C of a set C ∈ CF and a setA ∈ AF is again an element of the family CF .

Since AF consists of Borel sets, the σ-algebra σ(AF ) generated by AF is contained inB

d. It remains to prove the reversed inclusion: Bd ⊆ σ(AF ). This is done by showing

that each open set is a countable union of sets of AF .Let B be an open set and x ∈ B. There exists some r > 0 such that B(x, r) ⊂ B. Set

l := (diamO)−1 and letΣx =

w ∈ Σ(lr) : x ∈ SwO

.

By definition of Σ(lr), for all w ∈ Σx, diamSwO = rw diamO ≤ r and thus SwO ⊂B(x, r).

63

For each w ∈ Σ(lr) \Σx, SwO has some positive distance to x. Therefore we can findsome positive constant c such that d(x, SwO) > c for all w ∈ Σ(lr) \ Σx.

Let Cx = B(x, c) \⋃

w∈ΣxSwO, which is obviously a subset of O(lr)c and thus an ele-

ment of CF . Moreover, let Ax = Cx ∪⋃

w∈ΣxSwO. By construction, Ax is a finite union

of sets from AF and Ax ⊆ B(x, r) ⊂ B. On the other hand the family Ax : x ∈ Bcovers the set B and thus B =

⋃x∈B Ax. Since x ∈ intAx, the family intAx : x ∈ B

forms an open cover of B, which, by the Lindelof Theorem has an countable open sub-cover, i.e. there are x1, x2, . . . in B such that B =

⋃i intAxi

. But then also B =⋃

i Axi

and so, since each set Axiis a finite union of sets from AF , we obtain a representation

of B as a countable union of sets from AF , as desired. This completes the proof.

The properties derived in Lemma 5.1.1 are sufficient for AF to be a separating class forthe family of totally finite (signed) measures. We recall the uniqueness theorem whichis well known for positive measures and easily generalized to signed measures.

Theorem 5.1.2. Let µ and ν totally finite signed measures on Bd, and A an intersection

stable generator of Bd such that µ(A) = ν(A) for all A ∈ A. Then µ = ν.

Although we can not give a direct reference for this theorem for signed measures, webelieve that it is well known. For a proof of this statement for positive measures see forinstance Elstrodt [5, p.60] or Jacobs [16, p.51]. The arguments of the proof in [5] extendto signed measures, once one has verified the fact that for any increasing sequence ofsets An converging to A and any signed measure µ, µ(Ai) → µ(A) as n → ∞.

By Lemma 5.1.1, the family AF satisfies the hypotheses of Theorem 5.1.2 and is thusa separating class. AF is constructed in such a way that the measures considered caneasily be computed for the sets of this family.

In the proof of Theorem 1.4.1 we will have to show that certain measures coincidewith some multiple of the measure µF defined in (1.4.3). Since by the above uniquenesstheorem it is sufficient to compare the measures for sets A ∈ AF , we collect the valuesµF (A) for those sets. Recall that µF is the self-similar measure with weights rs

1, . . . , rsN,

i.e. it is the unique probability measure satisfying the invariance relation

µF =N∑

i=1

rsi µF S−1

i .

It is well known that, provided the OSC is satisfied, µF (O) = µF (F ) and similarlyµF (SwO) = µF (SwF ) for each w ∈ Σ∗. Since µF (F ) = 1, the invariance relation impliesµF (SwF ) = rs

w. Therefore, we record:

µF (SwO) = rsw for all w ∈ Σ∗ (5.1.1)

and

µF (C) = 0 for all C ∈ CF . (5.1.2)

64

5.2. Proof of Theorem 1.4.1 and 1.4.2

First we prove Theorem 1.4.1, for which we assume that F has polyconvex parallelsets. Fix some k ∈ 0, . . . , d and assume that the k-th scaling exponent of F satisfiessk = s − k. The first step is to deduce some technical estimates, which provide boundsfor the curvature measure of Fǫ in the sets (SwO)δ and SwO for w ∈ Σ∗. They are basedon the key lemma (Lemma 4.2.1). The exponent γ which will occur in all the estimatesbelow, is the one we defined in (4.1.9).

Lemma 5.2.1. Let w ∈ Σ∗. There exists c > 0 such that for all 0 < ǫ ≤ δ ≤ ρrw and• ∈ +,−, var

C•k(Fǫ, (SwO)δ) ≤ rk

wC•k(Fǫr−1

w) + cǫk−sδγ (5.2.1)

andC•

k(Fǫ, SwO) ≥ rkwC•

k(Fǫr−1w

) − cǫk−sδγ . (5.2.2)

Proof. Fix w ∈ Σ∗. Observe that (SwO)δ = (SwO)−δ ∪ (∂SwO)δ. Provided that ǫ ≤ δ,for the first set in this union, the locality of C•

k (here in the open set (SwO)−δ we have(SwF )ǫ ∩ (SwO)−δ = Fǫ ∩ (SwO)−δ) and the scaling property imply

C•k(Fǫ, (SwO)−δ) = C•

k((SwF )ǫ, (SwO)−δ)

= rkwC•

k(Fǫr−1w

, O−δr−1w

) (5.2.3)

≤ rkwC•

k(Fǫr−1w

).

Choosing r such that w ∈ Σ(r), i.e. rw < r ≤ rwr−1min, the second set is a subset of

(O(r)c)δ, since ∂SwO ⊆ O(r)c. Therefore, by Lemma 4.2.1 , there are constants c, γ > 0such that

C•k(Fǫ, (∂SwO)δ) ≤ C•

k(Fǫ, (O(r)c)δ) ≤ cǫk−sδγ

for all ǫ ≤ δ ≤ ρr (and so in particular for δ ≤ ρrw). Hence the first estimate (5.2.1)follows immediately from the relation

C•k(Fǫ, (SwO)δ) ≤ C•

k(Fǫ, (SwO)−δ) + C•k(Fǫ, (∂SwO)δ).

For the second estimate choose r such that O = O(r), i.e. 1 < r ≤ r−1min. Then, again by

Lemma 4.2.1, there are c′, γ > 0 such that for all ǫ ≤ δ ≤ ρr

C•k(Fǫ) ≤ C•

k(Fǫ, O−δ) + C•k(Fǫ, (O

c)δ)

≤ C•k(Fǫ, O−δ) + c′ǫk−sδγ .

Bringing c′ǫk−sδγ onto the other side of this inequality and taking into account (5.2.3)we infer that for all ǫ ≤ δ ≤ ρr

C•k(Fǫ, SwO) ≥ C•

k(Fǫ, (SwO)−δ)

= rkwC•

k(Fǫr−1w

, O−δr−1w

)

≥ rkw

(C•

k(Fǫr−1w

) − c′(ǫr−1w )k−s(δr−1

w )γ)

.

Hence the estimate (5.2.2) holds for the constant c = c′rs−γw for all ǫ ≤ δ ≤ ρr and thus

in particular for ǫ ≤ δ ≤ ρrw. For the maximum of the two constants c derived for thetwo estimates, both inequalities are satisfied, completing the proof of Lemma 5.2.1.

65

For convenience we introduce the abbreviation

ν(f) :=

∫

Rd

fdν (5.2.4)

for the integral of a function f with respect to a (signed) measure ν. For w ∈ Σ∗ andδ > 0, let fw

δ : Rd → [0, 1] be a continuous function such that

fwδ (x) = 1 for x ∈ SwO and fw

δ (x) = 0 for x outside (SwO)δ.

For simplicity, assume that fwδ ≤ fw

δ′ for all δ < δ′. Obviously, fwδ has compact support

and satisfies 1SwO ≤ fwδ ≤ 1(SwO)δ

. Moreover, as δ → 0, the functions fwδ converge

(pointwise) to 1SwO, implying in particular the convergence of the integrals ν(fwδ ) →

ν(1SwO) = ν(SwO) with respect to any (signed) Radon measure ν.

Now recall the definition of the rescaled curvature measures νk,ǫ from (1.4.1). Usingthe above estimates, we derive some bounds for the integrals νk,ǫ(f

wδ ).

Lemma 5.2.2. Let w ∈ Σ∗. Then for all 0 < ǫ ≤ δ ≤ ρrw

∣∣∣νk,ǫ(fwδ ) − rs

wνk,ǫr−1w

(Rd)∣∣∣ ≤ 2cδγ ,

where c = c(w) is the constant in Lemma 5.2.1.

Proof. Fix w ∈ Σ∗. Since 1SwO ≤ f δw ≤ 1(SwO)δ

, Lemma 5.2.1 implies that there existc, γ > 0 such that for all ǫ ≤ δ ≤ ρr

ν±k,ǫ(f

wδ ) ≤ ν±

k,ǫ((SwO)δ) ≤ rswν±

k,ǫr−1w

(Rd) + cδγ (5.2.5)

and similarly

ν±k,ǫ(f

wδ ) ≥ ν±

k,ǫ(SwO) ≥ rswν±

k,ǫr−1w

(Rd) − cδγ . (5.2.6)

Applying these inequalities to νk,ǫ(fwδ ) = ν+

k,ǫ(fwδ )− ν−

k,ǫ(fwδ ), we obtain that on the one

hand

νk,ǫ(fwδ ) ≤ rs

wν+

k,ǫr−1w

(Rd) + cδγ −(rswν−

k,ǫr−1w

(Rd) − cδγ)

= rswνk,ǫr−1

w(Rd) + 2cδγ

and on the other hand

νk,ǫ(fwδ ) ≥ rs

wν+

k,ǫr−1w

(Rd) − cδγ −(rswν−

k,ǫr−1w

(Rd) + cδγ)

= rswνk,ǫr−1

w(Rd) − 2cδγ .

Combining both estimates, the assertion of Lemma 5.2.2 follows immediately.

66

Proof of the convergence νk,ǫw−→ Cf

k (F ) µF . Assume that F is non-arithmetic.The total masses of the variation measures ν+

k,ǫ and ν−k,ǫ of νk,ǫ are uniformly bounded.

Moreover, since for each ǫ ≤ 1 the support of ν±k,ǫ is contained in the 1-parallel set of F ,

the families ν+k,ǫǫ∈(0,1] and ν−

k,ǫǫ∈(0,1] are tight. Therefore, by Prohorov’s Theorem,they are relatively compact, i.e. every sequence has a weakly convergent subsequence.In particular, every null sequence has a subsequence ǫn such that the measures ν+

k,ǫn

converge weakly, and this subsequence has a further subsequence, again denoted ǫn,such that also the measures ν−

k,ǫnconverge weakly.

Now let ǫn such a sequence, i.e. assume that

ν+k,ǫn

w−→ ν+k and ν−

k,ǫn

w−→ ν−k as n → ∞,

for some limit measures ν+k and ν−

k . The weak convergence of the variation measuresν±

k,ǫnimplies the weak convergence of the signed measures νk,ǫn = ν+

k,ǫn− ν−

k,ǫnand the

limit measure is given by νk = ν+k − ν−

k . Note that the measures ν+k and ν−

k are notnecessarily the positive and negative variation of νk. They are just some representationof νk as a difference of two positive measures and do not necessarily live on two disjointsets. In general, the limit measures νk might depend on the chosen sequence ǫn.However, it is our aim to show that here this is not the case, i.e. we want to prove thatfor any such sequence ǫn, the limit measure νk is the same, namely that νk coincides

with µk := Cfk (F ) µF , which implies at once that the weak limit νk,ǫ as ǫ → 0 exists and

coincides with µk, as stated in Theorem 1.4.1 for the non-arithmetic case.In Section 5.1 we constructed the family AF and showed that it is an intersection stable

generator of Bd. By Theorem 5.1.2, the measures νk and µk coincide if they coincide forall sets A ∈ AF . The measure µk is known. For sets C ∈ CF , by (5.1.2), µF (C) = 0 and

so µk(C) = Cfk (F )µF (C) = 0. For sets SwO, w ∈ Σ∗, by (5.1.1), µF (SwO) = rs

w and

thus µk(SwO) = Cfk (F )rs

w.Therefore, we have to show that for all w ∈ Σ∗

νk(SwO) = Cfk (F )rs

w, (5.2.7)

and for all C ∈ CF

νk(C) = 0. (5.2.8)

Proof of (5.2.7). We approximate the measure of the sets SwO with the integrals ofthe functions fw

δ defined in (5.2.4 and use Lemma 5.2.2. Fix w ∈ Σ∗ and let r = rw. ByLemma 5.2.2, we have for all n and δ such that ǫn ≤ δ ≤ ρr

∣∣∣νk,ǫn(fwδ ) − rs

wνk,ǫnr−1w

(Rd)∣∣∣ ≤ 2cδγ . (5.2.9)

Keeping δ fixed and letting n → ∞, the weak convergence implies νk,ǫn(fwδ ) → νk(f

wδ ),

since fwδ is continuous. Moreover, νk,ǫnr−1

w(Rd) = (ǫr−1

w )s−kCk(Fǫr−1w

) → Cfk (F ), by

Theorem 1.2.6. Hence the above inequality yields∣∣∣νk(f

wδ ) − rs

wCfk (F )

∣∣∣ ≤ 2cδγ (5.2.10)

for each δ ≤ ρr. Letting now δ → 0, the integrals νk(fwδ ) converge to νk(1SwO) =

νk(SwO), while the right hand side of the inequality vanishes. Therefore, |νk(SwO) −rswCf

k (F )| ≤ 0 which implies νk(SwO) = rswCf

k (F ), as claimed in (5.2.7).

67

Proof of (5.2.8). Fix r > 0. It suffices to show ν±k (O(r)c) = 0, since this immediately

implies that νk(C) = ν+k (C) − ν−

k (C) = 0 for all C ⊆ O(r)c. Similarly as before weapproximate the indicator function of O(r)c by continuous functions. For δ > 0, letgδ : R

d → [0, 1] a continuous function such that

gδ(x) = 1 for x ∈ O(r)c and gδ(x) = 0 for x ∈ (O(r))−δ. (5.2.11)

Since gδ ≤ 1(O(r)c)δ, by Lemma 4.2.1, for all ǫn ≤ δ ≤ ρr,

ν±k,ǫn

(gδ) ≤ cδγ .

Keeping δ fixed and letting n → ∞, the weak convergence implies that ν±k,ǫn

(gδ) →ν±

k (gδ) while the right hand side remains unchanged. Letting now δ → 0, the functionsgδ converge pointwise to 1O(r)c and thus ν±

k (gδ) → ν±k (O(r)c), while cδγ vanishes. Hence

ν±k (O(r)c) = 0, completing the proof of (5.2.8).

This completes the proof of the weak convergence of νk,ǫ to µk as ǫ → 0 for thenon-arithmetic case.

Now we want to discuss the weak convergence of the averaged measures νk,ǫ as definedin (1.4.2). For this purpose we first derive some estimate for the integrals νk,ǫ(f

wδ ), which

is the analogue to Lemma 5.2.2 for the averaged measures νk,ǫ.

Lemma 5.2.3. Let w ∈ Σ∗. For all ǫ and δ such that 0 < ǫ ≤ δ ≤ ρrw it holds

∣∣∣νk,ǫ(fwδ ) − rs

wνk,ǫr−1w

(Rd)∣∣∣ ≤ 2cδγ +

log δ

log ǫ2(cδγ + M),

where c = c(w) is the constant of Lemma 5.2.1 and M is the one of Lemma 1.2.2.

Proof. Fix w ∈ Σ∗. Since f δw ≤ 1(SwO)δ

,

ν±k,ǫ(f

wδ ) ≤ ν±

k,ǫ((SwO)δ) =1

|log ǫ|

∫ 1

ǫǫs−kC±

k (Fǫ, (SwO)δ)dǫ

ǫ

For ǫ ≤ δ, we split the integral into two parts, one over the interval [ǫ, δ] and one over(δ, 1), and apply the first inequality of Lemma 5.2.1 to the first part and Lemma 1.2.2to the second part. Then for all ǫ ≤ δ ≤ ρrw we obtain

ν±k,ǫ(f

wδ ) ≤ 1

|log ǫ|

∫ δ

ǫ(rs

wν±k,ǫr−1

w(Rd) + cδγ)

dǫ

ǫ+

1

|log ǫ|

∫ 1

δM

dǫ

ǫ

≤ rswν±

k,ǫr−1w

(Rd) +cδγ

|log ǫ|

∫ δ

ǫ

dǫ

ǫ+

M

|log ǫ|

∫ 1

δ

dǫ

ǫ

≤ rswν±

k,ǫr−1w

(Rd) + cδγ +log δ

log ǫ(cδγ + M). (5.2.12)

In a similar way we derive the corresponding lower bounds. Since 1SwO ≤ f δw, it holds

ν±k,ǫ(f

wδ ) ≥ ν±

k,ǫ(SwO) ≥ 1

|log ǫ|

∫ δ

ǫǫs−kC±

k (Fǫ, SwO)dǫ

ǫ.

68

Applying the second estimate of Lemma 5.2.1, we infer that for all ǫ ≤ δ ≤ ρrw

ν±k,ǫ(f

wδ ) ≥ 1

|log ǫ|

∫ δ

ǫ(rs

wν±k,ǫr−1

w(Rd) − cδγ)

dǫ

ǫ

≥ rswν±

k,ǫr−1w

(Rd) − 1

|log ǫ|

∫ 1

δrswν±

k,ǫr−1w

(Rd)dǫ

ǫ− cδγ

|log ǫ|

∫ δ

ǫ

dǫ

ǫ.

Since, by Lemma 1.2.2, ν±k,ǫr−1

w(Rd) ≤ M and rs

w ≤ 1, the second term is bounded from

below by − log δlog ǫM . Therefore we obtain

ν±k,ǫ(f

wδ ) ≥ rs

wν±k,ǫr−1

w(Rd) − cδγ − log δ

log ǫ(cδγ + M) (5.2.13)

Applying inequalities (5.2.12) and (5.2.13) to νk,ǫ(fwδ ) = ν+

k,ǫ(fwδ )−ν−

k,ǫ(fwδ ), the asserted

inequality follows in a similar way as the one for νk,ǫ(fwδ ) in the proof of Lemma 5.2.2.

Proof of the convergence νk,ǫw−→ C

fk(F ) µF . The proof for the averaged measures

νk,ǫ is almost the same as the one for νk,ǫ in the non-arithmetic case. It is easily seenthat also the families ν+

k,ǫǫ∈(0,1] and ν−k,ǫǫ∈(0,1] are tight and thus, by Prohorov’s

Theorem, relatively compact. Let ǫn be a null sequence such that

ν+k,ǫn

w−→ ν+k and ν−

k,ǫn

w−→ ν−k as ǫ → 0,

for some limit measures ν+k and ν−

k . Then νk,ǫn

w−→ νk := ν+k − ν−

k . Now we have to

show that the limit measure νk coincides with µk := Cfk(F ) µF , from which we can

conclude the asserted convergence as ǫ → 0.Again we work with the family AF . By Theorem 5.1.2, the measures νk and µk coincide

if they coincide for all sets A ∈ AF . Since, by (5.1.2), µk(C) = Cfk(F )µF (C) = 0 for

sets C ∈ CF and, by (5.1.1), µk(SwO) = Cfk(F )rs

w for w ∈ Σ∗, we have to show that forall w ∈ Σ∗

νk(SwO) = Cfk(F )rs

w, (5.2.14)

and for all C ∈ CF

νk(C) = 0 (5.2.15)

in order to complete the proof that νk,ǫw−→ C

fk(F ) µF .

Proof of (5.2.14). The arguments are very similar to those in the proof of (5.2.7). Fixw ∈ Σ∗ and let r = rw. Lemma 5.2.3 ensures that for all n and δ such that ǫn ≤ δ ≤ ρr

∣∣∣νk,ǫn(fwδ ) − rs

wνk,ǫnr−1w

(Rd)∣∣∣ ≤ 2cδγ +

log δ

log ǫn2(cδγ + M).

Keeping δ fixed and letting n → ∞, the weak convergence implies νk,ǫn(fwδ ) → νk(f

wδ ),

while νk,ǫnr−1w

(Rd) → Cfk(F ), by Theorem 1.2.6. On the right hand side the second term

vanishes. Hence the above inequality yields∣∣∣νk(f

wδ ) − rs

wCfk(F )

∣∣∣ ≤ 2cδγ

for each δ ≤ ρr. Letting now δ → 0, the integrals νk(fwδ ) converge to νk(SwO), while

the right hand side of the inequality vanishes, proving assertion (5.2.14).

69

Proof of (5.2.15). Fix r > 0. For the functions gδ defined in (5.2.11) there exists c > 0such that for all ǫ ≤ δ ≤ ρr,

ν±k,ǫ(gδ) ≤ cδγ +

log δ

log ǫ(cδγ + M),

which is derived in a similar way as (5.2.12).

Having established this estimate, the arguments are the same as those in the proofof (5.2.8). Set ǫ = ǫn in the above inequality and let first n → ∞ and afterwardsδ → 0. Then the right hand side converges to 0, while on the left hand side we end upwith ν±

k (O(r)), which therefore equals zero. Now the assertion (5.2.15) is an immediateconsequence.

This completes the proof of Theorem 1.4.1.

Proof of Theorem 1.4.2. Suppose now that F is an arbitrary self-similar set satisfyingOSC, i.e. the parallel sets Fǫ are not necessarily polyconvex. We will briefly outline, howthe arguments in the above proof have to be modified to obtain a proof of Theorem 1.4.2.

By Gatzouras’s theorem, the average Minkowski content M(F ) does always exist andis strictly positive implying that always sd = s − d. Going again through the proof ofTheorem 1.4.1, it is easily seen that for k = d most of the arguments remain valid inthe general case. Some parts even simplify due to the positivity of the measure. InLemma 5.2.1 we simply replace Cvar

d (Fǫ, · ) by λd(Fǫ ∩ · ) and use Lemma 4.6.2 insteadof Lemma 4.2.1 to obtain the corresponding general estimate. Lemma 5.2.2 and 5.2.3remain valid as stated, when the generalized definitions (1.4.4) and (1.4.5) of νd,ǫ andνd,ǫ are used. The proofs of both lemmas even simplify, since these measures are notsigned.

For proving the convergence νd,ǫw−→ M(F ) µF for non-arithmetic sets F , Prohorov’s

Theorem can now be applied directly to νd,ǫ, without switching to the signed measures

first. Choose any sequence ǫn such that νd,ǫn

w−→ νd and show that the limit measure νd

of this sequence equals µd := M(F ) µF and is thus independent of the chosen sequence.The equivalence of the measures is obtained with the same arguments as in in the proofof Theorem 1.4.1 taking into account Lemma 4.6.2 and the generalized Lemma 5.2.2.The proof of the convergence νd,ǫ

w−→ M(F ) µF is adapted in a similar way.

5.3. Proof of Theorem 1.4.3 and 1.4.4

For k = d − 1 and k = d, we show the weak convergence of the normalized curvaturemeasures ν1

k,ǫw−→ µF . First we discuss the case k = d− 1. The arguments for k = d are

very similar as we will briefly outline afterwards.

Proof of Theorem 1.4.3. Let F be a self-similar set as assumed in Theorem 1.4.3. Inparticular it was assumed that there exists a constant b > 0 such that

lim infǫ→0

ǫs−d+1Cd−1(Fǫ) = b, (5.3.1)

70

which immediately implies sd−1 = s − d + 1. By definition, for all ǫ > 0,

ν1d−1,ǫ = (ǫs−d+1Cd−1(Fǫ))

−1νd−1,ǫ.

If F is non-arithmetic, then the weak convergence ν1d−1,ǫ

w−→ µF follows immediately,

since, by the Theorems 1.2.6 and 1.4.1, ǫs−d+1Cd−1(Fǫ) converges to Cfd−1(F ), while

νd−1,ǫw−→ Cf

d−1(F ) µF . By the assumptions, Cfd−1(F ) > 0.

So assume now that F is arithmetic. In this case we have to work a bit more. Similarlyas in the above proofs, let ǫn a null sequence such that ν1

d−1,ǫn

w−→ ν1d−1 as n → ∞

for some limit measure ν1d−1. We prove that ν1

d−1 is independent of the choice of thesequence ǫn by showing that ν1

d−1 = µF . By Theorem 5.1.2 and the considerations inSection 5.1, it suffices to show that for all w ∈ Σ∗

ν1d−1(SwO) = rs

w, (5.3.2)

and for all C ∈ CF ,ν1

d−1(C) = 0. (5.3.3)

Proof of (5.3.2). Fix w ∈ Σ∗. Dividing the inequality in Lemma 5.2.2 by νd−1,ǫ(Rd), we

infer that for all ǫn ≤ δ ≤ ρrw

∣∣ν1d−1,ǫn

(fwδ ) − rs

wqw(ǫn)∣∣ ≤ 2cδγ(νd−1,ǫn(Rd))−1,

where

qw(ǫ) :=νd−1,ǫr−1

w(Rd)

νd−1,ǫ(Rd)=

(ǫr−1w )s−d+1Cd−1(Fǫr−1

w)

ǫs−d+1Cd−1(Fǫ).

We show that this quotient converges to 1 as ǫ → 0. Since F was assumed to bearithmetic, there is some h > 0 such that, for each i = 1, . . . , N , there exists ni ∈ N suchthat − log ri = nih. We infer from Remark 3.1.5 that the expression g(ǫ) = ǫs−d+1Ck(Fǫ)is asymptotic to some periodic function G(ǫ) of (multiplicative) period ζ = e−h. Notingthat rw = ζn is some integer power n of the period, we conclude that numerator anddenominatior of qw(ǫ) are asymptotic to the same function G(ǫr−1

w ) = G(ǫ). This impliesconvergence of the quotient qw(ǫ) to 1.

Letting now n → ∞ in the above inequality, the quotient qw(ǫn) converges to 1 whilethe right hand side is bounded from above by b−1, by the assumption (5.3.1). Hence

∣∣ν1d−1(f

wδ ) − rs

w

∣∣ ≤ 2cb−1δγ ,

and, by taking the limit δ → 0, we derive that ν1d−1(SwO) = rs

w, completing the proofof (5.3.2).

Proof of (5.3.3). It suffices to show that ν1d−1(O(r)c) = 0 for each r > 0. So fix some r.

Analogously to the proof of (5.2.8), we conclude from Lemma 4.2.1 that the functionsgδ (defined in (5.2.11)) satisfy the inequality

ν1d−1,ǫn

(gδ) =νd−1,ǫn(gδ)

νd−1,ǫn(Rd)≤ cδγ(νd−1,ǫn(Rd))−1

for all ǫn ≤ δ ≤ ρr. Hereby note that Cvard−1(Fǫ, · ) = Cd−1(Fǫ, · ). Now the assertion

easily follows by letting first n → ∞ and then δ → 0 and taking into account (5.3.1).

71

We have shown that for each null sequence ǫn, for which as n → ∞ the measuresν1

d−1,ǫnconverge at all, they converge to µF . Hence ν1

d−1,ǫw−→ µF as ǫ → 0, as we stated

in Theorem 1.4.3.

Proof of Theorem 1.4.4. Note that, by Gatzouras’s theorem, a condition similar to(5.3.1) is always satisfied: There exists some b > 0 such that lim infǫ→0 ǫs−dλd(Fǫ) = b.Hence there is no extra assumption required in this case. Now the arguments of theabove proof carry over to the case k = d and to arbitrary self-similar sets satisfyingOSC, when Theorem 1.2.10 and Lemma 4.6.2 are taken into account.

72

A. Appendix: Signed measures and weak

convergence

We summarize a few facts and definitions concerning signed measures, which we alwaysregard as totally finite signed measures here. In particular, we clarify the notion of weakconvergence of a sequence of signed measures. For more details on signed measures werefer to the text books on measure theory, e.g. the ones by Jurgen Elstrodt [5], JosephL. Doob [4] or Konrad Jacobs [16], and for the weak convergence of measures to PatrickBillingsley [3].

Signed measures. Let X a metric space and X the σ-algebra of Borel sets of X. Afunction µ : X → R is called a signed measure if it is σ-additive, i.e. for any sequence ofpairwise disjoint sets A1, A2, . . . ∈ X it holds

µ

(⋃

i

Ai

)=∑

i

µ(Ai).

In particular, this definition implies µ(∅) = 0 and |µ(X)| < ∞ . µ is called a measureor positive measure if µ(A) ≥ 0 for all A ∈ X .

We define the set functions µ+, µ− and µvar by setting for each A ∈ X

µ+(A) := supB⊆A

µ(B), µ−(A) := − infB⊆A

µ(B) and µvar(A) := µ+(A) + µ−(A) .

It can be shown that µ+, µ− and µvar are finite positive measures on X . They are calledrespectively the positive, negative and total variation measures of µ.

Theorem A.1.1. Let µ be a signed measure on a σ-algebra X . Then

(i) (Jordan decomposition) µ = µ+ − µ− .

(ii) (Hahn decomposition) X is the disjoint union of two sets X+, X− ∈ X such thatµ−(X+) = µ+(X−) = 0 . The sets X+ and X− are unique up to µvar-null sets.

Integration with respect to a signed measure. Recall that for a measurable functionf : X → R the integral with respect to a positive measure µ is defined as follows. Fornonnegative simple functions g =

∑ni=1 ci1Ai

, where Ai ⊆ X and ci > 0, set

∫

Xgdµ =

n∑

i=1

ciµ(Ai) .

73

Any nonnegative measurable function f is approximated from below by a sequenceg1, g2, . . . of simple functions and the integral is then defined as the limit

∫

Xfdµ = lim

j→∞

∫

Xgjdµ .

It can be shown that the limit does not depend on the choice of the sequence g1, g2, . . ..f is µ-integrable if the limit is finite. Finally, arbitrary measurable functions f are µ-integrable if the integrals

∫X f+dµ and

∫X f−dµ are both finite, and for such functions

f the integral is defined as∫

Xfdµ =

∫

Xf+dµ −

∫

Xf−dµ .

Here f+(x) = max f(x), 0 and f−(x) = f+(x)− f(x) denote the positive and negativepart of f , respectively.

The integral with respect to a signed measure µ is now defined with respect to itsJordan decomposition. A measurable function f : X → R is µ-integrable if and only ifit is µvar-integrable. Then the integral with respect to µ is defined as

∫

Xfdµ =

∫

Xfdµ+ −

∫

Xfdµ− .

Weak convergence of signed measures. Having defined the integral with respect to asigned measure, the generalization of the concept of weak convergence to signed measuresis straightforward. Let µ, µ1, µ2, . . . be signed measures on X . The sequence µn issaid to converge weakly to the limit measure µ, µn

w−→ µ as n → ∞, if

limn→∞

∫

Xfdµn =

∫

Xfdµ

for all bounded continuous functions f .It is obvious from the definition, that weak convergence of the variation measures µ+

n

and µ−n of µn to the variation measures µ+ and µ− of µ is sufficient for weak convergence

of a sequence of signed measures:

µ+n

w−→ µ+ and µ−n

w−→ µ− =⇒ µnw−→ µ

This implication suggests to investigate the variation measures, to which the theory ofweak convergence of (positive) measures applies, instead of studying the signed measuresthemselves. Note that the converse implication is not true. This is illustrated by a simpleexample.

Example A.1.2. Let x1, x2, . . . and y1, y2, . . . be two disjoint sequences in X, i.e. inparticular xn 6= yn for all n ∈ N, converging both to the same point x ∈ X. For eachn let µ+

n := δxn and µ−n := δyn the Dirac measures of xn and yn, respectively. Set

µn := µ+n − µ−

n . Obviously, µ±n

w−→ δx and therefore µnw−→ δx − δx = 0. Thus the limit

measure µ is the zero measure and so its positive and negative variations µ+ and µ− arezero as well. Hence we have µn

w−→ µ but not µ+n

w−→ µ+ and not µ−n

w−→ µ−.

74

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Erklarung

Ich erklare, daß ich die vorliegende Arbeit selbstandig und nur unter Verwendung

der angegebenen Quellen und Hilfsmittel angefertigt habe.

Jena, 29. Mai 2006

79