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- 1. Series in Mathematical Analysis and Applications Edited by
Ravi P. Agarwal and Donal ORegan VOLUME 9 NONLINEAR ANALYSIS 2005
by Taylor & Francis Group, LLC
- 2. SERIES IN MATHEMATICAL ANALYSIS AND APPLICATIONS Series in
Mathematical Analysis and Applications (SIMAA) is edited by Ravi P.
Agarwal, Florida Institute of Technology, USA and Donal ORegan,
National University of Ireland, Galway, Ireland. The series is
aimed at reporting on new developments in mathematical analysis and
applications of a high standard and or current interest. Each
volume in the series is devoted to a topic in analysis that has
been applied, or is potentially applicable, to the solutions of
scientific, engineering and social problems. Volume 1 Method of
Variation of Parameters for Dynamic Systems V. Lakshmikantham and
S.G. Deo Volume 2 Integral and Integrodifferential Equations:
Theory, Methods and Applications Edited by Ravi P. Agarwal and
Donal ORegan Volume 3 Theorems of Leray-Schauder Type and
Applications Donal ORegan and Radu Precup Volume 4 Set Valued
Mappings with Applications in Nonlinear Analysis Edited by Ravi P.
Agarwal and Donal ORegan Volume 5 Oscillation Theory for Second
Order Dynamic Equations Ravi P. Agarwal, Said R. Grace, and Donal
ORegan Volume 6 Theory of Fuzzy Differential Equations and
Inclusions V. Lakshmikantham and Ram N. Mohapatra Volume 7 Monotone
Flows and Rapid Convergence for Nonlinear Partial Differential
Equations V. Lakshmikantham, S. Koksal, and Raymond Bonnett Volume
8 Nonsmooth Critical Point Theory and Nonlinear Boundary Value
Problems Leszek Gasinski and Nikolaos S. Papageorgiou Volume 9
Nonlinear Analysis Leszek Gasinski and Nikolaos S. Papageorgiou
2005 by Taylor & Francis Group, LLC
- 3. Series in Mathematical Analysis and Applications Edited by
Ravi P. Agarwal and Donal ORegan VOLUME 9 NONLINEAR ANALYSIS Leszek
Gasinski Nikolaos S. Papageorgiou Boca Raton London New York
Singapore 2005 by Taylor & Francis Group, LLC
- 4. Published in 2005 by Chapman & Hall/CRC Taylor &
Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton,
FL 33487-2742 2005 by Taylor & Francis Group, LLC Chapman &
Hall/CRC is an imprint of Taylor & Francis Group No claim to
original U.S. Government works Printed in the United States of
America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International
Standard Book Number-10: 1-58488-484-3 (Hardcover) International
Standard Book Number-13: 978-1-58488-484-2 (Hardcover) Library of
Congress Card Number 2005045529 This book contains information
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Cataloging-in-Publication Data Gasinski, Leszek. Nonlinear analysis
/ Leszek Gasinski, Nikolaos S. Papageorgiou. p. cm. -- (Series in
mathematical analysis and applications ; v. 9) Includes
bibliographical references and index. ISBN 1-58488-484-3 1.
Nonlinear functional analysis. 2. Nonlinear operators. I.
Papageorgiou, Nikolaos Socrates. II. Title. III. Series.
QA321.5.G37 2005 515'.7--dc22 2005045529 Visit the Taylor &
Francis Web site at http://www.taylorandfrancis.com and the CRC
Press Web site at http://www.crcpress.com Taylor & Francis
Group is the Academic Division of T&F Informa plc. 2005 by
Taylor & Francis Group, LLC
- 5. To Prof. Zdzislaw Denkowski 2005 by Taylor & Francis
Group, LLC
- 6. Contents 1 Hausdor Measures and Capacity 1 1.1 Measure
Theoretical Background . . . . . . . . . . . . . . . . 3 1.2
Covering Results . . . . . . . . . . . . . . . . . . . . . . . . .
7 1.3 Hausdor Measure and Hausdor Dimension . . . . . . . . . 22
1.4 Dierentiation of Hausdor Measures . . . . . . . . . . . . . 44
1.5 Lipschitz Functions . . . . . . . . . . . . . . . . . . . . . .
. 52 1.6 Capacity . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 81 1.7 Remarks . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 103 2 Lebesgue-Bochner and Sobolev Spaces 107 2.1
Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . 108
2.2 Lebesgue-Bochner Spaces and Evolution Triples . . . . . . . 127
2.3 Compactness Results . . . . . . . . . . . . . . . . . . . . . .
150 2.4 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . .
. . . . 179 2.5 Inequalities and Embedding Theorems . . . . . . . .
. . . . . 213 2.6 Fine Properties of Functions and BV-Functions . .
. . . . . 239 2.7 Remarks . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 257 3 Nonlinear Operators and Young Measures 265
3.1 Compact and Fredholm Operators . . . . . . . . . . . . . . .
266 3.2 Operators of Monotone Type . . . . . . . . . . . . . . . .
. . 303 3.3 Accretive Operators and Semigroups of Operators . . . .
. . 343 3.4 The Nemytskii Operator and Integral Functions . . . . .
. . 405 3.5 Young Measures . . . . . . . . . . . . . . . . . . . .
. . . . . 427 3.6 Remarks . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 463 4 Smooth and Nonsmooth Analysis and Variational
Principles 467 4.1 Dierential Calculus in Banach Spaces . . . . . .
. . . . . . 468 4.2 Convex Functions . . . . . . . . . . . . . . .
. . . . . . . . . 488 4.3 Haar Null Sets and Locally Lipschitz
Functions . . . . . . . 501 4.4 Duality and Subdierentials . . . .
. . . . . . . . . . . . . . 512 4.5 Integral Functionals and
Subdierentials . . . . . . . . . . . 558 4.6 Variational Principles
. . . . . . . . . . . . . . . . . . . . . . 578 4.7 Remarks . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 599 vii 2005 by
Taylor & Francis Group, LLC
- 7. viii 5 Critical Point Theory 607 5.1 Deformation Results . .
. . . . . . . . . . . . . . . . . . . . . 608 5.2 Minimax Theorems
. . . . . . . . . . . . . . . . . . . . . . . 642 5.3 Structure of
the Critical Set . . . . . . . . . . . . . . . . . . 654 5.4
Multiple Critical Points . . . . . . . . . . . . . . . . . . . . .
661 5.5 Lusternik-Schnirelman Theory and Abstract Eigenvalue Prob-
lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 689 5.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 705 6 Eigenvalue Problems and Maximum Principles 707 6.1
Linear Elliptic Operators . . . . . . . . . . . . . . . . . . . .
708 6.2 The Partial p-Laplacian . . . . . . . . . . . . . . . . . .
. . . 732 6.3 The Ordinary p-Laplacian . . . . . . . . . . . . . .
. . . . . 759 6.4 Maximum Principles . . . . . . . . . . . . . . .
. . . . . . . . 775 6.5 Comparison Principles . . . . . . . . . . .
. . . . . . . . . . . 788 6.6 Remarks . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 797 7 Fixed Point Theory 803 7.1 Metric
Fixed Point Theory . . . . . . . . . . . . . . . . . . . 804 7.2
Topological Fixed Point Theory . . . . . . . . . . . . . . . . 821
7.3 Partial Order and Fixed Points . . . . . . . . . . . . . . . .
. 833 7.4 Fixed Points of Multifunctions . . . . . . . . . . . . .
. . . . 877 7.5 Remarks . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 891 Appendix 895 A.1 Topology . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 895 A.2 Measure Theory . . . . .
. . . . . . . . . . . . . . . . . . . . 899 A.3 Functional Analysis
. . . . . . . . . . . . . . . . . . . . . . . 908 A.4 Calculus and
Nonlinear Analysis . . . . . . . . . . . . . . . . 912 List of
Symbols 915 References 925 2005 by Taylor & Francis Group,
LLC
- 8. Preface Linear functional analysis deals with innite
dimensional topological vector spaces (which mix in a fruitful way
the linear (algebraic) structure with topo- logical one) and the
linear operators acting between them. The eort was to extend
standard results of linear analysis to an innite dimensional con-
text. The rst half of the twentieth century is marked by intensive
theoretical investigations in this area, which were also
accompanied by detailed treat- ment of linear mathematical models.
With the exception of a short period during the 1930's (compact
operators and Leray-Schauder degree), nonlin- ear operators were
out of the emerging picture. However, mounting evidence from
diverse other elds such as physics, engineering, economics, biology
and others suggested that there should be an eort to extend the
linear theory to various kinds of nonlinear operators. Systematic
eorts in this direction started in the early 1960's and mark the
beginning of what is known today as Nonlinear Analysis." Since then
several theories have been developed in this respect and today some
of them are well established approaching their limits, while others
are still the object of intense research activity. It is not a co-
incidence that simultaneously with the advent of nonlinear
analysis, we have the appearance of nonsmooth analysis and of
multivalued analysis, both of which were motivated by concrete
needs in applied areas such as control the- ory, optimization, game
theory and economics. Their development provided nonlinear analysis
with new concepts, tools and theories that enriched the subject
considerably. Today nonlinear analysis is a well established mathe-
matical discipline, which is characterized by a remarkable mixture
of analysis, topology and applications. It is exactly the fact that
the subject combines in a beautiful way these three items that
makes it attractive to mathematicians. The notions and techniques
of nonlinear analysis provide the appropriate tools to develop more
realistic and accurate models describing various phenomena. This
gives nonlinear analysis a rather interdisciplinary character.
Today the more theoretically inclined nonmathematician (engineer,
economist, biologist or chemist) needs a working knowledge of at
least a part of nonlinear analysis in order to be able to conduct a
complete qualitative analysis of his models. This supports a high
demand for books on nonlinear analysis. Of course the subject is
big (vast is maybe a more appropriate word) and no single book can
cover all its theoretical and applied parts. In this volume, we
have fo- cused on those topics of nonlinear analysis which are
pertinent to the theory of boundary value problems and their
applications such as control theory and calculus of variations. ix
2005 by Taylor & Francis Group, LLC
- 9. x In Chapter 1 we deal with Hausdor measures and capacities,
which provide the means to estimate the size" or dimension" of
thin" or highly irregular" sets. The recent development of fractal
geometry and its uses in a variety of applied areas (such as
Brownian motion of particles, turbulence in uids, geographical
coastlines and surfaces etc) renewed the interest on Hausdor
measures, which for a long period were a topic of secondary
importance within measure theory. In this chapter we also have our
rst encounter with Lipschitz At this point we prove the celebrated
Rademacher's theorem." Chapter 2 deals with certain classes of
function spaces, which arise naturally in the study of boundary
value problems. These are the Lebesgue-Bochner spaces (the suitable
spaces for the analysis of evolution equations) and the Sobolev
spaces (the suitable spaces for weak solutions of elliptic
equations). We conduct a detailed study of these spaces with
special emphasis on com- pactness and embedding results. Also using
the tool of Hausdor measures and capacities, we investigate the ne
properties of Sobolev functions and also introduce and study
functionals of bounded variation which are useful in theoretical
mechanics. In Chapter 3, we deal with certain large classes of
nonlinear operators which arise often in applications. We examine
compact operators for which we de- velop in parallel the
corresponding linear theory, with one of the main results being the
spectral theorem for compact self-adjoint operators on a Hilbert
space. We also investigate nonlinear operators of monotone type
which have their roots in the calculus of variations and exhibit
remarkable surjectivity properties. Monotone operators lead to
accretive operators, the two families being identical in the
context of Hilbert spaces. Accretive operators are closely
connected with the generation theory of semigroups of operators. We
also ex- amine both linear and nonlinear semigroups. Semigroups are
basic tools in the study of evolution equations. In addition, we
examine the Nemytskii operator which is a nonlinear operator
encountered in almost all problems. Finally, in the last section of
the chapter, we discuss Young measures which provide the right
framework to examine the limit behavior of the minimizing sequence
of variational problems which do not have a solution. Young
measures are used in optimal control and in the calculus of
variations in connection with the so-called relaxation method."
nonsmooth functions. We start with the G^ateaux and Frechet
derivatives. We discuss the generic dierentiability of continuous
convex functions (Mazur's theorem) and extend Rademacher's theorem
to locally Lipschitz functions between certain Banach spaces by
using the notion of Haar-null sets. Then we pass to nondierentiable
functions and develop the duality properties and subdierential
theory of convex functions and the generalized subdierential of
locally Lipschitz functions. We also examine integral functionals
and discuss the celebrated Ekeland variational principle
establishing its equivalence with some other geometric results of
nonlinear analysis. and locally Lipschitz functionals which will be
examined again in Chapter 4. Chapter 4 presents the calculus of
smooth and of certain broad classes of 2005 by Taylor & Francis
Group, LLC
- 10. xi In Chapter 5 we present the critical point theory of
C1-functions dened on a Banach space. This theory is in the core of
the variational methods used in the study of boundary value
problems. We follow the deformation approach which leads to minimax
characterizations of the critical values. We also study the
structure of the set of critical points and derive results on the
existence of multiple critical points. Next we present the
Lusternik-Schnirelman the- ory which extends to nonlinear
eigenvalue problems the corresponding linear theory of R. Courant.
Chapter 6 uses the abstract results of Chapter 5 as well as results
from earlier chapters to develop the spectrum of linear elliptic
dierential operators, of the partial p-Laplacian (with Dirichlet
and Neumann boundary conditions) and of the scalar and vector
ordinary p-Laplacian (with Dirichlet, Neumann and periodic boundary
conditions). We also present linear and nonlinear maximum
principles and comparison results, which are useful tools in the
study of boundary value problems. Finally in Chapter 7 we have
gathered some basic xed point theorems. We present results from
metric xed point theory, from topological xed point theory and xed
point results based on the partial order induced by a closed,
convex pointed cone. We also indicate how many of these results can
be extended to multifunctions (set-valued functions). We have tried
to make the volume self-contained. For this reason at the end the
general results used in the book. Nevertheless, within the test
whenever we are in the need of using some results not proved in the
book, we also give exact references where the interested reader can
nd additional information. Now that the project has reached its
conclusion, we would like to thank the good people of CRC Press
(especially Mrs. Jessica Vakili) for their help and kind
cooperation during the preparation of this book. We would like to
thank the two editors of this series, Prof. R.P. Agarwal and Prof.
D.O'Regan, for supporting this eort. of the book we have included a
rather extended appendix for easy reference of 2005 by Taylor &
Francis Group, LLC
- 11. Chapter 1 Hausdor Measures and Capacity During the golden
era of measure theory (namely the rst two decades of the 20th
century), Caratheodory was the rst to consider the notion of
length" for sets in RN. Later, in 1919, Hausdor, motivated by the
ideas of Caratheodory, introduced the measure and dimensional
concepts that we shall discuss in this chapter. So in the modern
language, the length" of a set A RN will be its Hausdor
one-dimensional outer measure (denoted by (1)). Following the
pioneering works of Caratheodory and Hausdor, signif- icant
contributions to the subject were made by Besicovitch. In fact, in
the rst decade of development of the subject, the main advances on
the subject were made by Besicovitch and his students, since
geometric measure theory was not part of the mainstream measure
theory. However, since the early 70's, the subject attracted a
large number of researchers, due to its fundamental importance in
the study of the so-called Fractal Geometry." Fractal sets arise in
many applications, such as turbulence in uids, geographical
coastlines and surfaces, uctuation of prices in stock exchanges,
the Brownian motion of par- ticles and others. Mandelbrojt was the
rst to emphasize their use to model a variety of phenomenona. There
have been many ways to estimate the size" or dimension" of small
(thin) sets and of highly irregular sets and to gen- eralize the
idea that points, curves and surfaces have dimensions 0, 1 and 2
respectively. Hausdor measure has the advantage of being a measure
and together with the notion of Hausdor dimension can provide a
more delicate sense of the size of sets in RN than Lebesgue measure
provides. To illustrate this, consider in R2 the set A df= t; sin 1
t : t 2 (0; 1) : Suppose we wish to measure the length of the curve
A. A rst approximation can be based on the Caratheodory outer
measure, which denes: 1(A) df= inf 1S n=1 A Cn 1X n=1 (Cn); i.e.,
the inmum is taken over all countable covers of A (by (A) we denote
the diameter of the set A; see (1.1)). If we adopt this denition,
we see that 1(A) < +1, while we know that the length of A is
innite. The reason for 1 2005 by Taylor & Francis Group,
LLC
- 12. 2 Nonlinear Analysis this is that in the denition of 1(A),
the covers of A are not forced to follow the geometry of A. For
this reason the Hausdor s-dimensional measures (s)(A) are dened as
limits of outer measures (s) which follow the local geometry of A
(see Denition 1.3.5). As another illustrative example, consider the
unit square S in R2 (i.e., square of side length equal to 1) and
dene 1(S) df= inf 1S n=1 S Cn 1X n=1 (Cn); i.e., again the inmum is
taken over all countable covers of S. We observe that we can do no
better than cover S itself. Indeed, if we cover S with smaller
squares of diameter less or equal to 1 n, then we see that we need
at least n2 squares to achieve the covering and so the
approximation of 1(S) obtained this way exceeds n p 2. So the
smaller the squares we use to cover, the bigger the estimate for
1(S). Therefore, small squares are irrelevant in the calculation of
1(S) and yet it is precisely them that should have an inuence on
the evaluation of 1(S). We expect 1(S) = 0, since the diameter is a
one-dimensional concept and it is used to measure a square in R2,
which is a two dimensional concept. For this we need a denition
which takes into account the local geometry of the set under
consideration. In this chapter, in Section 1.1 we recall some basic
denitions and facts from measure theory, which will be needed in
what follows. In Section 1.2, we discuss some covering theorems."
Covering results play a central role in geometric measure theory.
In Section 1.3 we introduce and study Hausdor measures and the
Hausdor dimension of sets. Among other things we calcu- late the
Hausdor dimension of some classical irregular sets in R
(Cantor-like sets). From these calculations, the reader will
realize that the Hausdor mea- sure and the Hausdor dimension of
sets (even of simple ones) may be hard to calculate. For this
reason sometimes other notions may be more suitable (such as
capacity; see Section 1.6). In Section 1.4 we discuss the
dierentia- tion of Hausdor measures and derive the
Lebesgue-Besicovitch dierentiation theorem. In Section 1.5, using
the tools of Hausdor measures, we study the geometry of Lipschitz
continuous functions. Among other things, we obtain the area and
coarea formulas" and the associated with them change of vari- ables
formulas." Finally in Section 1.6, we present an alternative
analytical notion measuring small sets in RN, namely the
p-capacity. We derive some basic properties of the p-capacities and
compare them to the Hausdor mea- sures. 2005 by Taylor &
Francis Group, LLC
- 13. 1. Hausdor Measures and Capacity 3 1.1 Measure Theoretical
Background In this section we recall some basic denitions and facts
from measure theory, which we shall need in the sequel. Let us
start with the concept of outer measure, which, when restricted to
a suitable -eld of sets, leads to a measure. DEFINITION 1.1.1 Let X
be a set. A map : 2X ! [0;+1] is said to be an outer measure, if
(a) (;) = 0; (b) A B =) (A) 6 (B) (monotonicity); (c) for any
sequence of sets fAngn>1 2X, we have 1[ n=1 An 6 1X n=1 (An)
(subadditivity). For a given outer measure on X and A 2 2X, we dene
the restriction of on A, denoted by bA, by (bA)(B) df= (A B) 8 B 2
2X: We say that is a nite outer measure if (X) < +1 (i.e., has
values in R+). REMARK 1.1.2 Note that bA is an outer measure on X,
while we dene jA to be the restriction of (as a function) on 2A,
i.e., jA : 2A ! [0;+1] is dened by jA (B) df= (B) 8 B 2 2A 2X:
Outer measures are useful because they lead to measures when
restricted to suitably dened -elds. These -elds can be quite large.
DEFINITION 1.1.3 Let X be a set and an outer measure on X. A set A
2 2X is said to be -measurable, if (B) = (A B) + (B n A) 8 B 2 2X;
i.e., A decomposes" every set B additively. 2005 by Taylor &
Francis Group, LLC
- 14. 4 Nonlinear Analysis REMARK 1.1.4 Let X be a set and an
outer measure on X. (a) By virtue of the subadditivity property of
an outer measure, to show that A 2 2X is -measurable, it is enough
to check that (B) > (A B) + (B n A) 8 B 2 2X: (b) Clearly, if A
2 2X and (A) = 0, then A is -measurable. (c) If A 2 2X, then any
-measurable set is also bA-measurable. (d) A is -measurable if and
only if Ac = X n A is -measurable. It is straightforward to check
the following result. PROPOSITION 1.1.5 If X is a set and is an
outer measure on X, then the collection of all -measurable sets is
a -eld and restricted on is a measure. REMARK 1.1.6 While Denition
1.1.3 involves only additivity of , the conclusion in Proposition
1.1.5 is about -additivity of on . This reveals the power of
Denition 1.1.3. Note that from Remark 1.1.4(b), it follows that the
-eld is -complete. DEFINITION 1.1.7 Let X be a nonempty Hausdor
topological space and let be an outer measure on X. (a) Let T be a
family of 2X. We say that is T -regular, if (A) = inf B 2 T A B (B)
8 A 2 2X: If T = , then we simply say that is regular. (b) We say
that is a Borel measure, if B(X) with B(X) being the Borel -eld of
X. (c) We say that is a Borel regular measure, if is a Borel
measure which is B(X)-regular. (d) We say that is a Radon measure,
if is a Borel regular measure and (K) < +1 8 K X; K-compact:
2005 by Taylor & Francis Group, LLC
- 15. 1. Hausdor Measures and Capacity 5 REMARK 1.1.8 Let X be a
Hausdor topological space and let be an outer measure on X. (a)
Note that is regular if and only if 8A 2 2X 9B 2 : (A) = (B): (b)
If is regular on X and fAngn>1 2X is increasing (i.e., An An+1
for n > 1), then 1[ n=1 An = sup n>1 (An): PROPOSITION 1.1.9
If X is a Hausdor topological space, is an outer measure on X which
is Borel regular and A 2 with (A) < +1, then bA is a Radon
measure. PROOF Let 1 df= bA: Evidently 1 and so 1 is a Borel
measure. Also for every compact K X, we have 1(K) < +1: It
remains to show that 1 is Borel regular. To this end note that
since is Borel regular, for a given A 2 2X, we can nd B 2 B(X), A
B, such that (A) = (B) < +1: Because A 2 , from Denition 1.1.3,
we have (B n A) = (B) (A) = 0: Since A 2 , for every C 2 2X, we
have (B C) n A (bB)(C) = (B C) = (B C A) + 6 (C A) + (B n A) = (C
A) = (bA)(C): As A B, we infer that bB = bA: So without any loss of
generality, we may assume that A 2 B(X). Let C 2 2X. Since is Borel
regular, we can nd D 2 B(X), such that A C D and (A C) = (D) 2005
by Taylor & Francis Group, LLC
- 16. 6 Nonlinear Analysis (see Remark 1.1.8(a)). Let us take E
df= D [ (X n A): Evidently E 2 B(X) and C (A C) [ (X n A) E:
Moreover, since E A = D A, we have 1(E) = (E A) = (D A) 6 (D) = (A
C) = 1(C); so 1 = bA is Borel regular (see Remark 1.1.8(a)), hence
Radon. We conclude this section, by recalling the following basic
measure theoretic approximations. PROPOSITION 1.1.10 If X is a
Hausdor topological space and is an outer measure on X which is
Borel, then (a) if A 2 B(X), (A) < +1 and " > 0, then we can
nd an open set U" A and a closed set C" A, such that (U" n C") <
"; i.e., (A) = inf U-open A U (U) = sup C-closed C A (C): (b) if is
Radon, then for every A 2 2X, we have (A) = inf U-open A U (U) and
if A 2 , then (A) = sup K-compact K A (K): REMARK 1.1.11 Note that
in the rst part of Proposition 1.1.10(b), the set A need not be
-measurable. 2005 by Taylor & Francis Group, LLC
- 17. 1. Hausdor Measures and Capacity 7 1.2 Covering Results One
of the main tools in geometric measure theory is the so called
Vitali covering theorem. For a given suciently large family of sets
that cover a given set A, Vitali's covering theorem allows us to
select a countable subfamily consisting of distinct sets with
exactly the desired approximation properties. The basic principle
embodied in the proof of Vitali's covering theorem is illustrated
in the next proposition. In what follows for any subset A of a
metric space (X; dX), we dene (A) = diam (A) df= sup x;y2A dX(x;
y); (1.1) the diameter of A (by convention diam ; df= 0).
PROPOSITION 1.2.1 If T is a collection of nondegenerate balls in RN
with sup B2T (B) < +1; then we can nd a nite or countable
subfamily F of T consisting of disjoint balls, such that [ B2T B [
B2F b B; with b B being the ball concentric with B, but with radius
ve times the radius of B. PROOF Let d0 df= sup B2T (B); Tn df= B 2
T : d0 2n < (B) 6 d0 2n1 8 n > 1: Inductively, we generate
subfamilies Fn Tn for n > 1. Namely, let F1 be any maximal
disjoint collection of balls in T1. Suppose we have selected F1; :
: : ;Fm. We choose Fm+1 to be any maximal disjoint subfamily of B 2
Tm+1 : B B0 = ; for all B0 2 m[ i=1 Fk and nally set F df= 1[ m=1
Fm: 2005 by Taylor & Francis Group, LLC
- 18. 8 Nonlinear Analysis Evidently F T and consists of disjoint
balls. Claim. For each B 2 T , we can nd B0 2 F, such that B B06= ;
and (B) 6 2(B0) (so also B B0). b For some m > 1, we have B 2
Tm. By virtue of the maximality of Fm, we mS can nd B0 k=1 Fk with
B B06= ;. We have that d0 2m 6 (B0) and (B) 6 d0 2m1 : So (B) 6
2(B0) and this proves the claim. From the claim it follows at once
that S B2T B S B02F b B0. DEFINITION 1.2.2 Let A RN. S A collection
T of sets in RN is said to be a Vitali cover of A, if A B2T B and
for every x 2 A and every " > 0, there exists B 2 T , such that
x 2 B and 0 < (B) < ". REMARK 1.2.3 Note that from the second
requirement of the above denition it follows that inf B2T (B) = 0:
So T is a Vitali cover of a set A, if every point x 2 A is
contained in an arbitrary small element of T . As a straightforward
consequence of Proposition 1.2.1 we obtain the follow- ing
proposition. PROPOSITION 1.2.4 If A RN, T is a Vitali cover of A
consisting of closed balls, such that sup B2T (B) < +1; then
there exists a countable family F = fBngn>1 consisting of
disjoint balls from T , such that for each m > 1, we have A m[
n=1 Bn [ 1[ n=m+1 b Bn; where b Bn is the closed ball cocentric
with Bn and radius ve times the radius of Bn. 2005 by Taylor &
Francis Group, LLC
- 19. 1. Hausdor Measures and Capacity 9 mn PROOF Let F be as in
the proof of Proposition 1.2.1. Select fBng =1 mS F. If A n=1 Bn,
then we are done. Otherwise let x 2 A n m[ n=1 Bn. Since T is a
Vitali cover of A consisting of closed balls, then we can nd B 2 T
, such that x 2 B and B Bn = ; 8 n 2 f1; : : : ;mg: But from the
claim in the proof of Proposition 1.2.1, we can nd B0 2 F, such
that B b B0 and B B06= ; (so B0 2 fBng1 n=m+1). Now we are ready to
state and prove Vitali's covering theorem. In what follows by N we
denote the N-dimensional Lebesgue outer measure. THEOREM 1.2.5
(Vitali Covering Theorem) If A RN with 0 < N(A) < +1 and T is
a Vitali cover of A consisting of closed sets, then we can nd a
sequence fCngn>1 of elements in T , such that CnCm = ; for n6= m
and N A n 1[ n=1 Cn = 0: PROOF Without any loss of generality, we
can assume that there exists an open set U RN with N(U) < +1 and
C U 8 C 2 T : We construct the sequence fCnginductively. Let C1 2 T
. Suppose that n>1 nS C1; : : : ;Cn are disjoint sets in T . If
A k=1 Ck, then we are nished. If not, setting Vn df= U n [n k=1 Ck;
we introduce Tn df= C 2 T : C Vn and n df= sup C2Tn N(C): Because A
n nS i=1 Ck6= ; and T is a Vitali cover of A, we see that Tn6= ;
and so n > 0. We select Cn+1 2 T with n 2 < N(Cn+1): 2005 by
Taylor & Francis Group, LLC
- 20. 10 Nonlinear Analysis We continue this process. Then either
at some nite step n > 1 we shall have nS A k=1 Ck, in which case
the proof of the theorem is complete or otherwise we produce a
sequence fCngn>1 T of disjoint sets. Then we have 1X n=1 N(Cn) =
N 1[ n=1 Cn 6 N(U) < +1: (1.2) For each n > 1 let Bn be a
ball with center in Cn and radius equal to 3(Cn). We claim that A n
[n k=1 Ck 1[ k=n+1 Bk 8 n > 1: (1.3) Let x 2 A n nS k=1 Ck.
Since T is a Vitali cover of A, we can nd a set Cx 2 Tn, such that
x 2 Cx and N(Cx) > 0: We shall show that Cx Ck6= ; for some k
> n: Indeed, if this is not the case, then N(Cx) 6 k for all k
> 1, which contra- dicts the fact that 0 6 lim k!+1 k 6 lim k!+1
2N(Ck+1) = 0 (recall the choice of Ck+1 and see (1.2)). Let m >
n be the smallest integer, such that Cx Cm6= ;: Since Cx 2 Tm1, we
have N(Cx) 6 m1 < 2N(Cm) and recalling the choice of Bm, also Cx
Bm. So we have proved (1.3). Then for any n > 1, we have N A n
1[ k=1 Ck 6 N A n [n k=1 Ck 6 1X k=n+1 N(Bk): (1.4) Recalling that
Bk is a ball of radius 3(Ck) and combining (1.2) and (1.4), we
conclude that N A n 1[ k=1 Ck = 0: 2005 by Taylor & Francis
Group, LLC
- 21. 1. Hausdor Measures and Capacity 11 Vitali's covering
theorem may be dicult to digest at rst and probably it is necessary
to see the lemma in action several times before appreciating it.
For this reason we present four simple applications from classical
analysis of functions of one-variable. We start with a denition
which establishes the notation for various limits of the dierence
quotient that we shall use in the sequel. These derivatives are
often more useful than the ordinary derivative, since they are
dened at every point. DEFINITION 1.2.6 For a given function f : [a;
b] ! R, the upper right and lower right derivates of f at x 2 [a;
b) are dened by D+f(x) df= lim sup h!0+ f(x + h) f(x) h and D+f(x)
df= lim inf h!0+ f(x + h) f(x) h respectively. Similarly the upper
left and lower left derivates of f at x 2 (a; b] are dened by Df(x)
df= lim sup h!0 f(x + h) f(x) h and Df(x) df= lim inf h!0 f(x + h)
f(x) h respectively. REMARK 1.2.7 Evidently, the derivates of a
function at a point may be innite. The function f is dierentiable
at x 2 (a; b), if 1 < D+f(x) = D+f(x) = Df(x) = Df(x) < +1:
The function f is dierentiable at x = a or at x = b, if the
appropriate two derivates are nite and equal. Also the one-sided
derivatives exist at a point x, if D+f(x) = D+f(x) and Df(x) =
Df(x): The derivates are also called Dini derivates and clearly we
always have D+f(x) 6 D+f(x) 8 x 2 [a; b) and Df(x) 6 Df(x) 8 x 2
(a; b]: 2005 by Taylor & Francis Group, LLC
- 22. 12 Nonlinear Analysis In the literature, sometimes we nd
the notion of a derived number for a function f at x. So 2 R is a
derived number for f at x, if there is a sequence fhngn>1 R;
such that hn ! 0; hn6= 0 8 n > 1 and lim n!+1 f(x + hn) f(x) hn
= : A function f may have many derived numbers at a point x. Of
course f is dierentiable at x if and only if all derived numbers of
f at x agree and are nite. EXAMPLE 1.2.8 Consider the function f :
R ! R dened by f(x) df= ( x sin 1 x if x6= 0; 0 if x = 0: We can
check that Df(0) = 1 < D+f(0) = 1 and every number in [1; 1] is
a derived number for f. The function f is not of bounded variation
(see Denition A.2.15(a)). LEMMA 1.2.9 If f : [a; b] ! R is
nondecreasing, then all four derivates of f are nite almost
everywhere on [a; b]. PROOF Clearly all derivates are nonnegative.
So it suces to show that D+f(x) < +1 and Df(x) < +1 for a.a.
x 2 [a; b]: Let A df= x 2 [a; b] : D+f(x) = +1 and suppose that (A)
= > 0; where is the Lebesgue outer measure on R. Let M > 0 be
such that f(b) f(a) < M 2 : 2005 by Taylor & Francis Group,
LLC
- 23. 1. Hausdor Measures and Capacity 13 For every x 2 A, we can
nd a decreasing sequence fhx ngn>1 with hx n & 0; hx n6= 0 8
n > 1; such that M 6 f(x + hx n) f(x) hx n : The collection [x;
x + hx n] x2A;n>1 is a Vitali cover of A. By virtue of Vitali's
covering theorem (see Theo- rem 1.2.5), we can nd a family of
disjoint intervals [xn; xn + hn] m n=1; such that Xm n=1 hn > 2
: Therefore Xm n=1 f(xn + hn) f(xn) > Xm n=1 Mhn > M 2 >
f(b) f(a); a contradiction. This proves that (A) = 0 and so D+f(x)
< +1: Analogously we can prove that Df(x) < +1. Using this
lemma and Vitali's covering theorem, we can now prove that a
nondecreasing function is dierentiable almost everywhere on [a; b].
THEOREM 1.2.10 If f : [a; b] ! R is nondecreasing, then f is
dierentiable almost everywhere on [a; b]. PROOF For f to be
dierentiable at x, we must have that all four derivates at x are
nite and equal. By virtue of Lemma 1.2.9, it suces to show that all
four derivates are equal almost everywhere. Let A df= x 2 (a; b) :
D+f(x) < D+f(x) : 2005 by Taylor & Francis Group, LLC
- 24. 14 Nonlinear Analysis We show that A is Lebesgue-null. The
proof for the other combinations of derivates is similar. Suppose
that (A) > 0: We can nd rational numbers r; s, such that the set
B df= x 2 A : D+f(x) < r < s < D+f(x) satises (B) = >
0: Let " 2 (0; ). From the regularity of the Lebesgue outer measure
, we know that there exists an open set U (a; b), such that B U and
1(U) " < : For each x 2 B and n > 1, we can nd hx n > 0;
with hx n & 0; such that x; x + hx n U and f(x + hx n) f(x) hx
n < r: The family [x; x + hx x2B;n>1 n] is a Vitali cover of
B. By virtue of Vitali's covering theorem (see Theo- rem 1.2.5),
for a given " > 0, we can nd a disjoint subfamily [xn; xn + hn]
m n=1 of the Vitali cover, such that B n m[ [xn; xn + hn] n=1 <
": We have Xm n=1 f(xn + hn) f(xn) < r Xm n=1 hn 6 r1(U) < r(
+ "): (1.5) Let us set C df= B m[ [xn; xn + hn] n=1 : We have that
" < (C): 2005 by Taylor & Francis Group, LLC
- 25. 1. Hausdor Measures and Capacity 15 For every y 2 C and k
> 1, we can nd uy k 2 y; y + 1 k , such that f(uy k) f(y) uy k y
> s and [y; uy k] (xn; xn + hn); for some n 2 f1; : : : ;mg: The
family [y; uy y2C;k>1 k] is a Vitali cover of C. Invoking
Vitali's covering theorem (see Theorem 1.2.5), we can nd a disjoint
subfamily [yk; uk] l k=1; such that (C) " < Xl k=1 (uk yk):
Hence, we have Xl (C) " k=1 f(uk) f(yk) > s Xl k=1 (uk yk) >
s > s( 2"): (1.6) For each 1 6 n 6 m, let Jn df= k 2 f1; : : : ;
lg : [yk; uk] (xn; xn + hn) : Since f is nondecreasing, using (1.6)
and (1.5), we have s( 2") < Xl k=1 f(uk) f(yk) = Xm n=1 X k2Jn
f(uk) f(yk) 6 Xm n=1 f(xn + hn) f(xn) < r( + "): Let " & 0,
to conclude that s 6 r, a contradiction. COROLLARY 1.2.11 If f :
[a; b] ! R is of bounded variation, then f is dierentiable almost
everywhere. PROOF Recall that f = f1 f2 with f1 and f2
nondecreasing and apply Theorem 1.2.10. 2005 by Taylor &
Francis Group, LLC
- 26. 16 Nonlinear Analysis THEOREM 1.2.12 If f : [a; b] ! R is
absolutely continuous (see Denition A.2.15(b)) and f0(x) = 0 for
a.a. x 2 [a; b]; then f is constant on [a; b]. PROOF We show that
f(u) = f(a) 8 u 2 (a; b]: So x a u 2 (a; b] and let A df= x 2 (a;
u) : f0(x) = 0 and "; > 0. Because f is absolutely continuous,
we can nd > 0, such that, if (rn; sn) m n=1 is a nite family of
disjoint subintervals of [a; b] with Xm n=m (sn rn) < ; then we
have Xm n=1 f(sn) f(rn) < ": We introduce the family T df= [x;
y] : x 2 A; x < y < u and f(y) f(x) y x < : Clearly T is a
Vitali cover of A. So by Vitali's covering theorem (see Theo- rem
1.2.5), we can nd a nite subfamily [xn; yn] m n=1 of disjoint sets
in T , such that 1 A n m[ [xn; yn] n=1 < : We may assume that a
< x1 < y1 < x2 < y2 < : : : < xm < ym < u:
Since 1 (a; u) n A = 0; we have 1 (a; u) n m[ [xn; yn] n=1 <
2005 by Taylor & Francis Group, LLC
- 27. 1. Hausdor Measures and Capacity 17 and 1 (a; u) n m[ [xn;
yn] n=1 = (x1 a) + Xm n=2 (xn yn1) + (u ym): Then, it follows that
f(u) f(a) 6 Xm n=1 f(yn) f(xn) + f(x1) f(a) + Xm n=2 f(xn) f(yn1) +
f(u) f(ym) < Xm (yn xn) + " 6 (u a) + ": n=1 Let "; & 0, to
obtain that f(u) = f(a). COROLLARY 1.2.13 If f : [a; b] ! R is
absolutely continuous, then f0 is Lebesgue integrable on [a; b] and
for all x 2 [a; b], we have f(x) f(a) = Zx a f0(s) ds: PROOF From
the fundamental theorem of Lebesgue calculus (see Theo- rem
A.2.20), we know that '(x) = Zx a f0(s) ds is absolutely continuous
on [a; b] and '0(t) = f0(t) for a.a. t 2 [a; b]: Then Theorem
1.2.12 implies that f = ' + , with a real constant . Evalu- ating
at x = a, we have f(a) = '(a) + = : Therefore Zx a f0(s) ds = f(x)
f(a) 8 x 2 [a; b]: 2005 by Taylor & Francis Group, LLC
- 28. 18 Nonlinear Analysis As a third application of Vitali's
covering theorem (see Theorem 1.2.5), we show that a very weak
local growth condition on a strictly increasing function f leads to
a strong local growth condition. More precisely, we prove the
following result. THEOREM 1.2.14 If f : [a; b] ! R is a strictly
increasing function, r > 0, A [a; b] and at each point x 2 A
there exists a derived number (see Remark 1.2.7), such that < r,
then f(A) 6 r(A). PROOF If (A) = +1, then the inequality is
obvious. So assume that (A) < +1: For a given " > 0, we can
nd a bounded open set U R, such that A U and 1(U) " < (A): If x
2 A, then by hypothesis we can nd a sequence fhngn>1 R n f0g,
such that hn ! 0; [x; x + hn] U 8 n > 1 (or [x + hn; x] U in the
event hn < 0; but in the sequel for simplicity we shall write
[x; x + hn] for both cases) and f(x + hn) f(x) hn < r 8 n >
1: (1.7) For all n > 1 and x 2 A, let Dn(x) df= [x; x + hn];
En(x) df= f(x); f(x + hn) : Because f is strictly increasing En(x)
is a nondegenerate, closed interval and f Dn(x) En(x) 8 n > 1; x
2 A: Since 1 Dn(x) = jhnj and 1 En(x) = f(x + hn) f(x) ; from
(1.7), we have 1 En(x) < r1 : (1.8) Dn(x) 2005 by Taylor &
Francis Group, LLC
- 29. 1. Hausdor Measures and Capacity 19 Passing to the limit as
n ! +1, we have jhnj ! 0 and so from (1.8), we obtain that lim n!+1
1 = 0: En(x) Let T df= En(x) x2A;n>1: Then T is a Vitali cover
of the set f(A). So Vitali's covering theorem (see Theorem 1.2.5)
implies the existence a disjoint sequence Enk (xk) k>1 T ; such
that 1 f(A) n 1[ k=1 Enk (xk) = 0: (1.9) Using (1.9) and (1.8), it
follows that f(A) 6 1 1[ k=1 Enk (xk) = 1X k=1 1 (Enk (xk)) < r
1X k=1 1 : (1.10) Dnk (xk) Since f is strictly increasing, we see
that Dnk (xk) k>1 are pairwise disjoint too. So we have 1X k=1 1
Dnk (xk) = 1 1[ k=1 Dnk (xk) (1.11) 6 1(U) 6 (A) + ": (1.12) From
(1.10) and (1.11), we infer that f(A) 6 r (A) + " : Let " & 0,
to conclude that f(A) 6 r(A): In a similar fashion, we can have the
following comparison result. 2005 by Taylor & Francis Group,
LLC
- 30. 20 Nonlinear Analysis THEOREM 1.2.15 If f : [a; b] ! R is a
strictly increasing function, s > 0, A [a; b] and at each point
x 2 A there exists a derived number , such that > s, then >
s(A): f(A) The nal application of Vitali's covering theorem (see
Theorem 1.2.5) is the following criterion for measurability of sets
in R. THEOREM 1.2.16 If F is any collection of intervals in R and A
= [ D2F D; then A is Lebesgue measurable. PROOF Let T be a
collection of all intervals E; such that E D for some D 2 F.
Evidently T is a Vitali cover of A and so by Vitali's covering
theorem (see Theorem 1.2.5), we can nd a se- quence fEngn>1 of
disjoint elements in T , such that A n 1[ n=1 En = 0: Because each
En A, the set A df= 1[ n=1 En [ A n 1[ n=1 En is Lebesgue
measurable. REMARK 1.2.17 Theorem 1.2.16 can be used to show that
the upper and lower derivates of an arbitrary function are
measurable. In particular then the four derivates of a measurable
function are measurable and so is the derivative of a measurable
function. We will not go into that here. 2005 by Taylor &
Francis Group, LLC
- 31. 1. Hausdor Measures and Capacity 21 When N is replaced by
an arbitrary Radon measure on RN, there is no systematic way to
control ( b B) in terms of (B). So the proof of Vitali's covering
theorem (see Theorem 1.2.5) which uses the principle involved in
Proposition 1.2.1, namely the use of suitable expansions of balls,
does not work. So we need an analog of Proposition 1.2.1, which
does not require enlarging the balls, though. This is done by the
so-called Besicovitch covering theorem." THEOREM 1.2.18
(Besicovitch Covering Theorem) If F is any collection of closed
balls in RN, sup B2F (B) < +1 and A is the set of centers of all
balls B 2 F; then there exist a positive integer k = k(N) > 1
and Tn F 8 n 2 f1; : : : ; kg; such that each Tn is a countable
collection of disjoint balls in F and A [k n=1 [ B2Tn B: Using the
above theorem, we can have the following counterpart of Vitali's
covering theorem (see Theorem 1.2.5). THEOREM 1.2.19 If is a Borel
measure on RN, T is a family of nondegenerate closed balls in RN, A
is the set of centers of balls in T ; (A) < +1, inf Br(a)2F r =
0 8 a 2 A and U RN is an open set, then there exists a countable
collection of disjoint balls F from T , such that [ B2F (A U) n B U
and [ B2F B = 0: 2005 by Taylor & Francis Group, LLC
- 32. 22 Nonlinear Analysis 1.3 Hausdor Measure and Hausdor
Dimension Hausdor measures were introduced as certain lower
dimensional measures on RN which allow us to measure small" subsets
in RN. The Hausdor measure and the associated Hausdor dimension of
the set provide a more delicate sense of the size of a set in RN
than the Lebesgue measure provides. We start with the introduction
of a special class of outer measures, known as metric outer
measures. DEFINITION 1.3.1 Let (X; dX) be a metric space (d is the
metric function). (a) If A;B X, then we say that A and B are
separated sets, if dX(A;B) df= inf a 2 A b 2 B dX(a; b) > 0: (b)
If is an outer measure on X, then we say that is a metric outer
measure, if (A [ B) = (A) + (B) 8 A;B X; A and B separated: We show
that if is a metric outer measure, then B(X) (), i.e., is Borel. To
this end we need the following auxiliary result, known as
Caratheodory's lemma. In what follows (X; d) is a metric space.
LEMMA 1.3.2 (Caratheodory Lemma) If is a metric outer measure on X,
U X is an open subset, U6= X, A U and An df= x 2 A : d(x;Uc) > 1
n 8 n > 1; (1.13) then (A) = lim n!+1 (An). PROOF Note that the
sequence fAngn>1 is an increasing sequence and so lim n!+1 (An)
exists. Moreover, since An A for n > 1, we have lim n!+1 (An) 6
(A): So we need to show that (A) 6 lim n!+1 (An): (1.14) 2005 by
Taylor & Francis Group, LLC
- 33. 1. Hausdor Measures and Capacity 23 Because U is open, we
have d(x;Uc) > 0 8 x 2 A and so we can nd n0 > 1 large enough
so that x 2 An0 . Therefore, we have A = 1[ n=1 An: For each n >
1, we introduce the set Cn df= An+1 n An = x 2 A : 1 n + 1 6
d(x;Uc) < 1 n : We have A = A2n [ 1[ k=2n Ck = A2n [ 1[ k=n C2k
[ 1[ k=n C2k+1 and from the subadditivity of , it follows that (A)
6 (A2n) + 1X k=n (C2k) + 1X k=n (C2k+1): (1.15) If both series are
convergent, then we obtain (1.14). So suppose that this is not true
and, say, we have 1X k=1 (C2k) = +1: (1.16) Note that d C2k;C2k+2
> 1 2k + 1 1 2k + 2 8 k > 1 and so the sets fCkgk>1 are
separated. Therefore, we have n[1 k=1 C2k = nX1 k=1 (C2k) 8 n >
1: (1.17) Note that n[1 k=1 C2k A2n 8 n > 1 and so n[1 k=1 C2k 6
(A2n) 8 n > 1: (1.18) 2005 by Taylor & Francis Group,
LLC
- 34. 24 Nonlinear Analysis From (1.17) and (1.18), it follows
that nX1 k=1 (C2k) 6 (A2n): Combining this with (1.16), we infer
that lim n!+1 (A2n) = +1 and so (A) 6 lim n!+1 (A2n); as desired.
Similarly, if 1P k=1 (C2k+1) = +1. THEOREM 1.3.3 If is an outer
measure on X, then B(X) () (i.e., is Borel) if and only if is a
metric outer measure. PROOF =)": Let A1;A2 X be separated sets and
let us set df= d(A1;A2) > 0: For every x 2 A1, we dene U(x) df=
B2 (x) = y 2 X : d(y; x) < 2 and U df= [ x2A1 U(x): Evidently U
is open, A1 U and A2 U = ;. Since by hypothesis U 2 (), we have
that (A1 [ A2) = (A1 [ A2) U + (A1 [ A2) Uc : (1.19) Because A1 U
and A2 U = ;, from (1.19), it follows that (A1 [ A2) = (A1) + (A2);
i.e., is metric outer measure. f= (=": It suces to show that ()
contains all closed sets. So let C X be closed and let us set dU
Cc. Let D X, A df= D n C and let fAngn>1 be an increasing
sequence of subsets of A as in Lemma 1.3.2. Then d(An;C) > 1 n 8
n > 1 2005 by Taylor & Francis Group, LLC
- 35. 1. Hausdor Measures and Capacity 25 and, from Lemma 1.3.2,
we have (D n C) = (A) = lim n!+1 (An): (1.20) Since by hypothesis
is a metric outer measure and the sets fAngn>1 are separated
from C, we have (D C) [ An (D) > = (D C) + (An) 8 n > 1:
Passing to the limit as n ! +1 and using (1.20), we obtain (D) >
(D C) + (D n C): The reverse inequality is always true
(subadditivity). So we obtain (D) = (D [ C) + (D n C) 8 D X: Thus C
2 () and hence B(X) (). To introduce the concept of Hausdor
measure, we shall need the following notion. Recall that by (X; d)
we denote a metric space. DEFINITION 1.3.4 A sequence fAngn>1 of
subsets of X is a -cover of a set C, if C 1[ n=1 An and (An) 6 8 n
> 1: By T(C) we denote the family of all -covers of the set C.
Using this notion, we can introduce the Hausdor s-dimensional
measure, s > 0. As usual, for any A X, (A) = diam (A) df= sup
x;y2A d(x; y); the diameter of A (by convention diam ; df= 0).
DEFINITION 1.3.5 For any s > 0, 0 < 6 +1 and C X, we dene (s)
(C) df= inf fAngn>12T(C) 1X n=1 (An)s (as always we use the
convention that inf ; = +1). The Hausdor s- dimensional outer
measure (s) is dened by (s)(C) df= lim &0 (s) (C) = sup >0
(s) (C): 2005 by Taylor & Francis Group, LLC
- 36. 26 Nonlinear Analysis REMARK 1.3.6 It is easily seen that
(s) is an outer measure. More- over, it is a metric outer measure.
Indeed, if > 0 is less than the positive distance of two
separate sets A and C, then no set in T(A[ C) can intersect both A
and C and so it follows that (A) + (s) (s) (A [ C) = (s) (C):
Letting & 0, we can obtain the same equality for (s). In
addition by Theorem 1.3.3, (s) is Borel. The restriction of (s) on
(s) is called the Hausdor s-dimensional measure. Sometimes it is
convenient to consider -covers consisting of open or alternatively
closed sets. In these cases, although a dierent value of (s) may be
attained for > 0, the limit (s) as (s) is dierent, if & 0 is
the same (see Davies (1970)). However, the limit we restrict
ourselves to -covers by balls (see Besicovitch (1928)). In this
case the resulting Hausdor measure is called the spherical Hausdor
measure. Finally, if X = RN, it is easy to see that (s) remains the
same if we consider -covers consisting only of convex sets. Next we
show that for any set C X, there is a critical value s0, such that
for s > s0, the corresponding Hausdor s-dimensional measure of C
is zero, while for s < s0 the Hausdor s-dimensional measure of C
is innite. THEOREM 1.3.7 If A RN and 0 6 s < t < +1, then (a)
if (s)(A) < +1, then (t)(A) = 0; (b) if (t)(A) > 0, then
(s)(A) = +1. PROOF (a) Let (s)(A) < +1 and t > s. Let
fAngn>1 2 T 1 m (A). Then for any n > 1, we have (An)t (An)s
= (An)ts 6 1 m ts ; so (t) 1 m (A) 6 1X n=1 (An)t 6 1 m ts 1X n=1
(An)s and thus (t) 1 t (A) 6 1 m ts (s) 1 m (A): Letting m ! +1, we
obtain (t)(A) = 0. (b) Let (t)(A) > 0 and s < t. Assuming
that (s)(A) < +1, from (a), we get that (t)(A) = 0, a contra-
diction. 2005 by Taylor & Francis Group, LLC
- 37. 1. Hausdor Measures and Capacity 27 f= This theorem leads
to the following denition. DEFINITION 1.3.8 Let C X. If there is no
s > 0, such that (s)(C) = +1, then dimC d0. Otherwise, let dimC
df= sup s > 0 s: (s)(C) = +1 Then dimC is called the Hausdor
dimension of C. Consider the Cantor ternary set C. It is well known
that C is a nonempty, bounded, nowhere dense, perfect set in R
which has Lebesgue measure zero. So the Lebesgue measure can
contribute no additional information concerning the size of C. On
the other hand, as we shall see the Hausdor dimension provides a
more delicate sense of size. PROPOSITION 1.3.9 If C [0; 1] is the
Cantor ternary set, then dimC = ln 2 ln 3 . PROOF We start with two
simple observations concerning the Hausdor s-dimensional outer
measure (s) on R. First note that (s) is translation invariant,
namely (s)(A) = (s)(A + x) 8 A R; x 2 R (here A+x df= a+x : a 2 A
). Second, (s) is s-positive homogeneous, i.e., for every # > 0,
(s)(#A) = #s(s)(A) 8 # > 0: In the construction of C we start by
removing from [0; 1] the open middle third 1 3 ; 2 3 . The
resulting set consists of two closed intervals 0; 1 3 and 2 3 ; 1 .
Let C1 df= C 0; 1 3 and C2 df= C 2 3; 1 : Evidently C1 and C2 are
translates of a multiple (by 1 3 ) of C. So we have (s)(C) = (s)(C1
[ C2) = (s)(C1) + (s)(C2) = 2(s) C2 = 2 1 3 s (s)(C) (1.21) (see
Remark 1.3.6 and the observations in the beginning of this proof).
From (1.21), it follows that (s)(C) = 0 or (s)(C) = +1 or 2 1 3 s =
1: 2005 by Taylor & Francis Group, LLC
- 38. 28 Nonlinear Analysis From the last possibility, it follows
that s = ln 2 ln 3: If we can show that 0 < (s)(C) < +1, then
s = ln 2 ln 3 is the Hausdor dimension of C (see Theorem 1.3.7).
First we show that (s)(C) > 0. Note that d(C1;C2) > 1 3: Let
6 1 3 . Then any collection fAngn>1 2 T(C) (which can be taken
to consist of open intervals; see Remark 1.3.6) can be decomposed
into two subcollections of intervals fAn;1gn>1 2 T(C1) and
fAn;2gn>1 2 T(C2), such that 1X n=1 (An)s = 1X n=1 (An;1)s + 1X
n=1 (An;2)s: (1.22) In the right hand side of (1.22) suppose that
the rst sum is smaller than the second. Because C2 is a translate
of C1, the same translation when applied to the intervals
fAn;1ggives a subcollection A0 n>1 n;1 n>1 2 T(C2). Also from
fAn;1gn>1 we can produce in a similar way a collection fA0
ngn>1 covering C, such that (A0 n) = 3(A0 n;1) 8 n > 1:
(1.23) Then, from (1.23) and the choice of s, we have 1X n=1 (An)s
> 1X n=1 (An;1)s + 1X n=1 (A0 n;1)s = 2 1X n=1 (A0 n;1)s = 2 1X
n=1 1 3 s (A0 n)s = 1X n=1 (A0 n)s: If any one of the intervals fA0
ngn>1 has length bigger or equal to 1 3 , we have 1X n=1 (An)s
> 1 3 s = 1 2: Because C is compact, we can use only nite
coverings and so min n>1 (An) > 0: The intervals fA0 ngn>1
are multiples (by (1.23)) of a subfamily of the intervals
fAngn>1, hence we have 3 min n>1 (A0 n) > min n>1 (An):
2005 by Taylor & Francis Group, LLC
- 39. 1. Hausdor Measures and Capacity 29 If every interval A0 n
has length (diameter) less than 1 3 , we can apply the same process
to the cover fA0 ngn>1. After a nite number of such steps, we
produce a cover fA00 ngn>1, such that max n>1 (A00 n) > 1
3 and 1X n=1 (An)s > 1X n=1 (A00 n)s; so 1X n=1 (An)s > 1 3 s
= 1 2 and thus 0 < (s)(C): Next we show that (s)(C) < +1: Let
fAngn>1 2 T(C) consist of open intervals. From this family, as
above, we obtain covers fAn;kgn>1 of Ck for k 2 f1; 2g, such
that (An;k) 6 3 8 n > 1: Again from the choice of s, we have
(An)s = (An;1)s + (An;2)s; so (s) (C) > (s) 3 (C): Because (s)
is nondecreasing in > 0, we infer that (s) is independent of
> 0. So we can take an open interval of length greater than 1 as
an open cover of C and conclude that (s)(C) 6 1. This proves that
dimE = ln 2 ln 3: One can show that for every 2 [0; 1], there
exists a set A R, such that dimA = . This can be done using
Cantor-like sets. These are sets which share most of the properties
of the Cantor ternary set, but need not be Lebesgue-null. We can
construct a Cantor-like set as follows. We start with the interval
[0; 1] and proceed inductively. We remove an open interval B1;1
centered at 1 2 with length less than 1. We are left with closed
intervals 2005 by Taylor & Francis Group, LLC
- 40. 30 Nonlinear Analysis D1;1 and D1;2 each with length less
than 1 2 . At the n-th step of this process we are left with closed
intervals Dn;1;Dn;2; : : : ;Dn;2n each with length less than 1 2n .
In the (n + 1)-st step, from each closed interval Dn;k we remove an
open interval En+1;k having the same center as Dn;k and length less
than the length of Dn;k. We set Sn df= 2n [ k=1 Dn;k and S df= 1
n=1 Sn: The set S is a Cantor-like set. Cantor ternary set).
However, unlike the Cantor ternary set, S need not be
Lebesgue-null. More precisely, consider a sequence f#ngn>1 of
positive numbers, such that 1 > 2#1 > 4#2 > : : : >
2n#n > : : : : Following the construction of S above, we remove
from [0; 1] an open interval centered at 1 2 and having length
12#1. The remaining closed intervals D1;1 and D1;2 each have length
#1. Then from each of the intervals D1;1 and D1;2 we remove
cocentric open intervals each of length #1 2#2. We are left with
closed intervals D2;1, D2;2, D2;3 and D2;4 each of length #2. We
continue this way. In the n-th step we are left with 2n closed
intervals each with length #n. Then we have 1(S) = lim n!+1 2n#n (1
being the Lebesgue measure on R). If #n = 1 3n , then S = C is the
Cantor ternary set. Although S is nowhere dense, we can have 1(S)
as close to 1 as we choose. Indeed, for a given 2 (0; 1), let #n
df= 1 2n n + 1 n + 1 8 n > 1: Then we have 1(S) = . Suppose that
in the construction of the Cantor-like set at each step the closed
subintervals are divided in the same proportions as the original,
namely (D1;1) = (D1;2) = # (D2;1) = (D2;2) = (D2;3) = (D2;4) = #2
and in general (Dn;k) = #k 8 k 2 f1; : : : ; 2ng: Then the
resulting Cantor-like set is denoted by S#. Arguing as in the proof
of Proposition 1.3.9, we obtain the following Proposition. 2005 by
Taylor & Francis Group, LLC It is known (see Hewitt &
Stromberg (1975, p. 71)) that S is nonempty, compact, nowhere dense
and perfect (just as the
- 41. 1. Hausdor Measures and Capacity 31 PROPOSITION 1.3.10 If #
2 0; 1 2 , then dim S# = ln 2 ln #. REMARK 1.3.11 If # = 1 3 , then
S = C is the Cantor ternary set and Propositions 1.3.9 and 1.3.10
coincide. COROLLARY 1.3.12 For each 2 [0; 1], there exists A R,
such that dimA = . PROOF If = 0, then we take A to be a singleton.
If 0 < < 1, then take # = exp ln 2 < 1 2 and use
Proposition 1.3.10. If = 1, let A = I = [0; 1]. Then we can easily
check that (s)(A) = 8< : +1 if 0 < s < 1; 1 if s = 1; 0 if
s > 1: Therefore dimA = 1. REMARK 1.3.13 An alternative way to
dene the Hausdor dimension of a set A X is by dimA df= inf s > 0
s: (s)(A) = 0 In general the Hausdor dimension of a set may be any
number in [0;+1] and need not be an integer. Even if dimA is an
integer and k = dimA > 0, the set A need not be a k-dimensional
surface" in any sense (see Federer (1969)). Next we turn our
attention to the case X = RN. Let us begin by recalling the
denition of the N-dimensional outer measure N. DEFINITION 1.3.14
(a) We say that Q RN is a closed N-cube, NQ if there exist ak <
bk for k = 1; : : : ;N, such that Q = [ak; bk]. We set k=1 jQj df=
NY (bk ak): k=1 (b) The Lebesgue N-dimensional outer measure N, for
all A RN, is dened by N(A) df= inf 1X k=1 jQkj : A 1[ k=1 : Qk; Qk
is closed N-cube 2005 by Taylor & Francis Group, LLC
- 42. 32 Nonlinear Analysis REMARK 1.3.15 Clearly the denitions
of 1 and (1) on R coincide. We shall show that for any N > 1 the
outer measures N and (N) are closely related. In fact they dier by
a multiplicative constant. This is not easy to establish and
requires some preparation which culminates to the so-called
isodiametric inequality," which says that the set of maximal volume
for a given diameter is the sphere. LEMMA 1.3.16 If f : RN ! [0;+1]
is Lebesgue measurable, then the set H df= (x; #) 2 RN R : 0 6 # 6
f(x) is Lebesgue measurable in RN+1. PROOF Let A df= x 2 RN : f(x)
= +1 : Then A is Lebesgue measurable. Let g : Ac R+ ! R+ be dened
by g(x; #) df= f(x) # 8 (x; #) 2 Ac R+: Evidently g is a
Caratheodory function (i.e., it is Lebesgue measurable in x 2 RN
and continuous in # 2 R). Therefore g is Lebesgue measurable on Ac
R+ and so H0 df= (x; #) 2 Ac R+ : # 6 f(x) is Lebesgue measurable
in RN+1. Finally note that H = H0 [ (A R+): In what follows for a;
b 2 RN, kakRN = 1, we introduce the following objects: L(a; b) df=
b + ta : t 2 R - the line passing from b in the direction of a and
P(a) df= x 2 RN : (x; a)RN = 0 - the plane passing from the origin,
perpendicular to a. 2005 by Taylor & Francis Group, LLC
- 43. 1. Hausdor Measures and Capacity 33 DEFINITION 1.3.17 Let a
2 RN with kakRN = 1 and A RN. We dene the Steiner symmetrization of
A with respect to the plane P(a) to be the set S(a;A) df= [ b 2
P(a) A L(a; b)6= ; b + ta : jtj 6 1 2(1) A L(a; b) : REMARK 1.3.18
The above dened Steiner symmetrization with re- spect to an (N
1)-dimensional subspace Y of RN is the operation which associates
to each A RN, the set V RN, such that for every L perpen- dicular
to Y either L A = ; and L V = ;; or L A6= ; and L V is a closed
segment centered in Y and (1)(L A) = (1)(L V ): If A is compact,
then V is compact too and N(A) = N(V ): Also if A is convex, then V
is convex too. The next Proposition summarizes the properties of
the Steiner symmetriza- tion. PROPOSITION 1.3.19 Let A RN and a 2
RN. (a) S(a;A) 6 (A). (b) If A RN is Lebesgue measurable, then so
is S(a;A) and N S(a;A) = N(A). PROOF (a) Assume that (A) < +1 or
otherwise the result is trivial. Also we may assume that A is
closed. For a given " > 0, let x; y 2 S(a;A) be such that S(a;A)
" 6 kx ykRN : Let b df= x (x; a)RN a and c df= y (y; a)RN a: 2005
by Taylor & Francis Group, LLC
- 44. 34 Nonlinear Analysis Then b; c 2 P(a). Let us set r df=
inf t 2 R : b + ta 2 A ; s df= sup t 2 R : b + ta 2 A ; u df= inf t
2 R : c + ta 2 A ; v df= sup t 2 R : c + ta 2 A : We may assume
that without any loss of generality that v r > s u: So v r >
1 2 (v r) + 1 2 (s u) = 1 2 (s r) + 1 2 (v u) > 1 2(1) + A L(a;
b) 1 2(1) : A L(a; c) Note that (x; a)RN 6 1 2(1) A L(a; b) and (y;
a)RN 6 1 2(1) A L(a; c) (recall that x; y 2 S(a; b)). It follows
that v r > (x; a)RN + (y; a)RN > (x y; a)RN : Hence we have
S(a;A) " 2 6 kx yk2 RN 6 kb ck2 RN + (x y; a)RN 2 6 kb ck2 RN + (v
r)2 = 2 (b + ra) (c + va) RN 6 (A)2 (note that A is closed and so b
+ ra; c + va 2 A). It follows that S(a;A) " 6 (A): Let " & 0,
to conclude that S(a;A) 6 (A). (b) Recall that the Lebesgue measure
N is rotation invariant. So we may take a = eN = 2 6664 0...01 3
7775 : 2005 by Taylor & Francis Group, LLC
- 45. 1. Hausdor Measures and Capacity 35 Then P(a) = P(eN) =
RN1: Note that the function f : RN1 ! R, dened by f(b) df= (1) A
L(a; b) 8 b 2 RN1; is measurable (Fubini's theorem) and N(A) = Z A
f(b)dN1(b) (since 1 = (1); see Remark 1.3.15). So by virtue of
Lemma 1.3.16, we have that S(a; b) df= (b; #) 2 RN1 R : f(b) 2 6 #
6 f(b) 2 n (b; 0) 2 RN1 R : A L(a; b) = ; is Lebesgue measurable in
RN and, moreover, N S(a;A) = Z RN1 f(b) dN1(b) = N(A): Now we are
properly equipped to prove the so-called isodiametric inequal-
ity," which states that, if in RN we consider the family of all
sets with given diameter, the one with maximum Lebesgue
N-dimensional outer measure (N- volume) is the sphere. THEOREM
1.3.20 (Isodiametric Inequality) For all A RN, we have (A) N(A) 6
a(N) 2 N ; where a(N) df= N2 N 2 ! is the volume of the unit ball
in RN. PROOF If (A) = +1, then there is nothing to prove. So
suppose that (A) < +1: 2005 by Taylor & Francis Group,
LLC
- 46. 36 Nonlinear Analysis Let fekgNk =1 be the standard basis
of RN. We introduce A1 = S(e1;A); A2 = S(e2;A1); : : : ; AN =
S(eN;AN1): Let us set A = AN. Claim 1. A is symmetric with respect
to the origin. By virtue of the denition of the Steiner
symmetrization, we have that A1 is symmetric with respect to the
plain P(e1). Let 1 6 k 6 N 1 and suppose that Ak is symmetric with
respect to P(e1); : : : ; P(ek). Again Ak+1 is symmetric with
respect to P(ek+1). Let us x 1 6 m 6 k and let Rm: RN ! RN be
reection with respect to P(em). Let b 2 P(ek+1). Because Rm(Ak) =
Ak, we have (1) Ak L(ek+1; b) = (1) Ak L(ek+1;Rm(b)) ; so t 2 R : b
+ tek+1 2 Ak+1 = t 2 R : Rm(b) + tek+1 2 Ak+1 and thus Rm(Ak+1) =
Ak+1; i.e., Ak+1 is symmetric with respect to P(em). It follows
that A = AN is symmetric with respect to P(e1); : : : ; P(eN),
hence it is symmetric with respect to the origin. Claim 2. N(A) 6
N2 N 2 ! (A) 2 N . Let x 2 A. Then because of Claim 1, we have x 2
A and so 2 kxkRN 6 (A): Hence A B(A) 2 (0) = y 2 RN : kykRN 6 (A) 2
and so N(A) 6 N B(A) 2 (0) 6 N2 N 2 ! (A) 2 N : Using Claim 2, we
can have the isodiametric inequality. Note that A RN is Lebesgue
measurable and so by Proposition 1.3.19, we have N A = N A and A 6
A : 2005 by Taylor & Francis Group, LLC
- 47. 1. Hausdor Measures and Capacity 37 Using Claim 2, it
follows that N(A) 6 N = N A A 6 N2 N 2 ! (A ) 2 N 6 N2 N 2 ! (A) 2
N = N2 N 2 ! (A) 2 N : THEOREM 1.3.21 If A RN, then N(A) =
cN(N)(A), with cN df= N2 2N N 2 ! . PROOF For a given " > 0, we
can nd a cover fCngn>1 of A consisting of closed, convex sets,
such that 1X n=1 (Cn)N 6 (N)(A) + ": By virtue of Theorem 1.3.20,
we have N(Cn) 6 cN(Cn)N 8 n > 1: So N(A) 6 1X n=1 N(Cn) 6 cN 1X
n=1 (Cn)N 6 cN(N)(A) + cN": Let " & 0 to conclude that N(A) 6
cN(N)(A): (1.24) To prove the opposite inequality, rst we show that
(N) is absolutely contin- uous with respect to N (see Denition
A.2.22). Note that for any N-cube Q, we have N(Q) = jQj 6 (Q) p N N
: So for a given > 0, we have (N) (A) 6 inf Qn -N-cube A 1S n=1
Qn (Qn) 6 1X n=1 (Qn) 6 p N N N(A): 2005 by Taylor & Francis
Group, LLC
- 48. 38 Nonlinear Analysis Let & 0, to conclude that (N) is
absolutely continuous with respect to N (see Denition A.2.22). Next
for a given "; > 0, we can nd a cover fQngn>1 of A consisting
of N-cubes, such that (Qn) < 8 n > 1 and 1X n=1 N(Qn) 6 N(A)
+ ": (1.25) We may suppose that N-cubes are open by expanding them
slightly so that the above inequality remains valid. Invoking
Vitali's covering theorem (see Theorem 1.2.5), for every n > 1
we can nd disjoint balls fBn;kgk>1 contained in Qn, such that
(Bn;k) 6 and N Qn n 1[ k=1 Bn;k = 0: By virtue of the absolute
continuity of (N) with respect to N, we have (N) Qn n 1[ k=1 Bn;k =
0 and (N) Qn n 1[ k=1 Bn;k = 0: Therefore, using (1.25), we have
(N) (A) 6 1X k=1 (N) (Qn) 6 1X n=1 1X k=1 (N) (Bn;k) + 1X n=1 (N)
Qn n 1[ k=1 Bn;k 6 1X n=1 1X k=1 (Bn;k)N = 1X n=1 1X k=1 1 cN
N(Bn;k) 6 1 cN 1X n=1 N(Qn) 6 1 cN N(A) + " cN : Let "; & 0, to
conclude that cN(N)(A) 6 N(A): (1.26) From (1.24) and (1.26), we
conclude that N = cN(N): 2005 by Taylor & Francis Group,
LLC
- 49. 1. Hausdor Measures and Capacity 39 REMARK 1.3.22 Some
authors, in order to get rid of the multiplicative constant cN,
normalize the denition of the Hausdor measures on RN. So if C RN, 0
6 s < +1, 0 < 6 +1, they set (s) (C) df= inf C 1S n=1 An (An)
6 1X n=1 (An) a(s) 2 s ; where a(s) df= s2 ( s 2 + 1) . Here (s)
df= Z+1 0 xs1ex dx is the gamma Euler function. The Hausdor
s-dimensional outer measure (s) is dened by (s)(C) = lim &0 (s)
(C) = sup >0 (s) (C) (cf., e.g., Evans & Gariepy (1992, p.
60)) . Recall that N B(x; r) = a(N)rN 8 x 2 RN: In this case
Theorem 1.3.21 says that N = (N): Note that (0) is the counting
measure. Let us prove some further properties of the Hausdor
measures on RN. PROPOSITION 1.3.23 Let 0 6 s < +1. We have (a)
(s)(A) = 0 for all A RN and all s > N. (b) (s)(A) = s(s)(A) for
all A RN and all > 0. (c) (s) K(A) = (s)(A) for all A RN and for
any ane isometry K: RN ! RN. PROOF (a) Let Q = (0; 1)N and let m
> 1 be an integer. For k = (ki)Ni =1 2 K df= f0; : : : ;m 1gN;
we set Qk df= NY i=1 ki m ; ki + 1 m : 2005 by Taylor & Francis
Group, LLC
- 50. 40 Nonlinear Analysis Note that Q = [ k2K Qk and (Qk) = p N
m : So we have (s) p N m (Q) 6 X k2K (Qk)s = mNs p N s : Letting m
! +1, since s > N, we obtain (s)(Q) = 0; from which it follows
that (s)(RN) = 0: (b) Note that for all C RN, we have (C) = (C): So
the result follows at once from Denition 1.3.5. (c) Note that for
all C RN, we have K(C) = (C): Again the result follows from
Denition 1.3.5. The next Proposition suggests a convenient way to
check that (s) vanishes on a set. PROPOSITION 1.3.24 If A RN, 0
< 6 +1 and 0 6 s < +1 are such that (s) (A) = 0, then (s)(A)
= 0. PROOF If s = 0, then (0) (A) = 0 implies that A = ; and so
(0)(A) = 0. So suppose that s > 0. For a given " > 0, we can
nd fCngn>1, such that A 1[ n=1 Cn; (Cn) 6 and 1X n=1 (Cn)s 6 ":
Evidently (Cn)s 6 " 8 n > 1 and so (s) " (A) 6 ". Let " & 0,
to conclude that (s)(A) = 0: 2005 by Taylor & Francis Group,
LLC
- 51. 1. Hausdor Measures and Capacity 41 Taking into account
that for a Lipschitz continuous function with constant c > 0,
for every A RN, we have f(A) 6 c(A); and we obtain the following
result. PROPOSITION 1.3.25 If f : RN ! RM is a Lipschitz continuous
function with Lipschitz constant c > 0 (see Denition 1.5.1), A
RN and 0 6 s < +1, then (s) f(A) 6 cs(s)(A). We conclude this
section by returning to the notion of Hausdor dimension (see
Denition 1.3.8) and having a second look at this concept. The
Hausdor dimension has an intuitive appeal when familiar objects are
under consideration. So for example dimRN = N (see Theorem 1.3.21).
Suppose we want to determine the Hausdor dimension of a curve C R3.
Our rst guess will be that dimC = 1. But recall that there are
curves in R3 which ll the unit cube. Such a curve must have Hausdor
dimension 3. Therefore we must proceed with caution. DEFINITION
1.3.26 Let (X; d) be a metric space. (a) By a curve in X we mean
the image f [0; 1] of a continuous function f : [0; 1] ! X. (b) The
length of a curve C = f [0; 1] is dened by l(C) df= sup Xm k=1 d
f(xk1); f(xk) ; where the supremum is taken over all partitions 0 =
x0 < x1 < : : : < xm = 1 of [0; 1]: (c) The curve C is
said to be rectiable, if l(C) < +1. REMARK 1.3.27 A curve C is a
continuum, i.e., a compact and con- nected set in X. In particular
then a curve is a Borel set; hence it is also (s)-measurable.
Moreover, if in Denition 1.3.26(a) f is injective, then f1 exists
and is continuous and so C is the homeomorphic image of [0; 1].
Also in Denition 1.3.26(a), we can replace [0; 1] by any closed
bounded interval [a; b]. Some authors require f to be injective.
2005 by Taylor & Francis Group, LLC
- 52. 42 Nonlinear Analysis PROPOSITION 1.3.28 If (X; d) is a
metric space, f : [0; 1] ! X is a nonconstant curve with length l
and C = f [0; 1] , then (a) 0 < (1)(C) 6 l; (b) if f is
injective, then (1)(C) = l. Therefore, if l is rectiable (i.e., l
< +1), then dimC = 1. PROOF (a) First we show that (1)(C) 6 l:
Assume that l < +1 or otherwise there is nothing to prove. Let
fAkgmk =1 be a collection of closed subarcs of C, such that C = m[
k=1 Ak; (Ak) 6 1 n and (1) 1 n (C) 6 Xm k=1 (Ak): (1.27) Let us
explicitly construct the subarcs Ak for k 2 f1; : : : ;mg. Note
that f is uniformly continuous and so we can nd > 0, such that d
f(x); f(y) < 1 n 8 x; y 2 [0; 1]; jx yj < : Consider a
partition 0 = x0 < x1 < : : : < xn = 1 of [0; 1]; such
that jxk xk1j < 8 k 2 1; : : : ;m : Let Ak df= f [xk1; xk] ; 8 k
2 1; : : : ;m : Evidently the subarcs fAkgmk =1 cover C and d
f(xk1); f(xk) 6 (Ak) < 1 n 8 k 2 f1; : : : ;mg: Note that every
Ak is compact and so we can nd points yk; zk 2 [xk1; xk], yk 6 zk,
such that d f(yk); f(zk) = (Ak): We generate the ner partition 0 6
y1 6 z1 6 y2 6 z2 6 : : : 6 ym 6 zm 6 1: 2005 by Taylor &
Francis Group, LLC
- 53. 1. Hausdor Measures and Capacity 43 From (1.27), we have
(1) 1 n (C) 6 Xm k=1 (Ak) = Xm k=1 d f(yk); f(zk) 6 l: Passing to
the limit as n ! +1, we obtain that (1)(C) 6 l. Next we show that 0
< (1)(C). To this end note that if 0 6 a < b 6 1, then d
f(a); f(b) 6 (1) : (1.28) f([a; b]) To see this let h: E df= f([a;
b]) ! R be the function h(u) df= d u; f(a) : Evidently h is a
Lipschitz continuous function with Lipschitz constant 1 and J df=
0; h(b) = 0; d f(a); f(b) h(E): So, from Proposition 1.3.25, we
have d f(a); f(b) = 1(J) = (1)(J) 6 (1) 6 (1)(E): h(E) This proves
inequality (1.28). But from (1.28) and since for appropriately
chosen a; b we have d f(a); f(b) > 0 (recall that the curve is
nonconstant), we conclude that 0 < (1)(C). (b) Now suppose that
f is injective. Let 0 = x0 < x1 < : : : < xm = 1 be a
partition of [0; 1]. The sets Ak df= f [xk1; xk] are pairwise
disjoint Borel subsets of X. Using inequality (1.28) on each
subarc, we obtain Xm k=1 d f(xk1); f(xk) 6 Xm k=1 (1) f [xk1; xk] =
(1) m[ k=1 f [xk1; xk] = (1) f [0; 1] = (1)(C): Since the partition
of [0; 1] was arbitrary, it follows that l 6 (1)(C). Com- bining
this with (a), we obtain that l = (1)(C). 2005 by Taylor &
Francis Group, LLC
- 54. 44 Nonlinear Analysis 1.4 Dierentiation of Hausdor Measures
From the general measure theory, we know that the dierentiation
theory of real functions can be extended to a theory of
dierentiation for measures, which has many similar features and
interesting problems. For the Lebesgue measures N, N > 1, one of
the basic results of this theory is the so-called Lebesgue density
theorem, which we recall here. THEOREM 1.4.1 (Lebesgue Density
Theorem) If A RN is a Lebesgue measurable set, then lim r&0
N(Br(x) A) N(Br(x)) = 8< : 1 for N-a.a. x 2 A; 0 for N-a.a. x 2
RN n A: DEFINITION 1.4.2 Let A RN and x 2 RN. We say that: (a) x is
a point of density of A, if lim r&0 N(Br(x) A) N(Br(x)) = 1;
(b) x is a point of dispersion of A, if lim r&0 N(Br(x) A)
N(Br(x)) = 0: REMARK 1.4.3 According to Theorem 1.4.1, we see that
N-almost every point of A is a point of density of A and N-almost
every point of RN nA is a point of dispersion of A. We can think
that the point of density of a set A form a kind of measure
theoretic interior of A, while the points of dispersion of A form a
kind of measure theoretic exterior of A. The purpose of this
section is to establish analogs of Theorem 1.4.1 for lower
dimensional Hausdor measures. In what follows we work in RN and 1
< s < N. THEOREM 1.4.4 If A RN is (s)-measurable and (s)(A)
< +1, then lim r&0 (s)(Br(x) A) (2r)s = 0 for (s)-a.a. x 2
RN n A: 2005 by Taylor & Francis Group, LLC
- 55. 1. Hausdor Measures and Capacity 45 PROOF For every t >
0, let Ct df= x 2 RN n A : lim sup r&0 (s)(Br(x) A) : (2r)s
> t To nish the proof it is enough to show that (s)(Ct) = 0 8 t
> 0: Fix " > 0. We know that (s)bA is a Radon measure (see
Proposition 1.1.9). So we can nd K A compact, such that (s)(A n K)
6 " (see Proposition 1.1.10(b)). Let U df= RN n K: Then U is open
and Ct U: For xed > 0, we consider the family of closed balls T
df= Br(x) : Br(x) U; 0 < r < ; (s)(Br(x) A) (2r)s > t :
Without any loss of generality we may assume that T6= ; or
otherwise Ct = ; and so (s)(Ct) = 0: Invoking Proposition 1.2.1, we
can nd a sequence Brn(xn) n>1 of disjoint elements in T , such
that Ct 1[ n=1 B5rn(xn): Then we have (s) 10(Ct) 6 1X n=1 (10rn)s 6
5s t 1X n=1 (s) Brn(xn) A 6 5s t (s)(U A) = 5s t (s)(A n K) 6 5s" t
: Let & 0, to obtain (s)(Ct) 6 5s" t : Since " > 0 was
arbitrary, we conclude that (s)(Ct) = 0. 2005 by Taylor &
Francis Group, LLC
- 56. 46 Nonlinear Analysis To have a complete analog of Theorem
1.4.1, we need to check and see if something can be said about the
density of A at its points. To do this we will make use of
Proposition 1.2.4. THEOREM 1.4.5 If A RN is (s)-measurable and
(s)(A) < +1, then 1 2s 6 lim sup r&0 (s)(Br(x) A) (2r)s 6 1
for (s)-a.a. x 2 A: PROOF First we show that lim sup r&0
(s)(Br(x) A) (2r)s 6 1 for (s)-a.a. x 2 A: (1.29) To this end, for
every t > 1, we introduce the set Ct A dened by Ct df= x 2 A :
lim sup r&0 (s)(Br(x) A) (2r)s > t : Fix " > 0. Again
(s)bA is a Radon measure (see Proposition 1.1.9). We can nd an open
set U RN, such that Ct U and (s)(U A) " 6 (s)(Ct) (1.30) (see
Proposition 1.1.10(b)). We introduce the family T of closed balls
dened by T df= Br(x) : Br(x) U; 0 < r < ; (s)(Br(x) A) (2r)s
> t : By virtue of Proposition 1.2.4, we can nd a sequence
Brn(xn) n>1 of dis- joint balls in T , such that Ct m[ n=1
Brn(xn) [ 1[ n=m+1 B5rn(xn) 8 m > 1: Then for > 0, we have
(s) 10(Ct) 6 Xm n=1 (2rn)s + 1X n=m+1 (10rn)s 6 1 t Xm n=1 (s) +
Brn A 5s t 1X n=m+1 (s) Brn(xn) A 6 1 t (s)(U A) + 5s t (s) 1[
n=m+1 Brn(xn) A 8 m > 1: 2005 by Taylor & Francis Group,
LLC
- 57. 1. Hausdor Measures and Capacity 47 Using (1.30) and
letting m ! +1, we obtain (s) 10(Ct) 6 1 t (s)(U A) 6 1 t (s)(Ct) +
" : Letting & 0, we see that (s)(Ct) 6 1 t (s)(Ct) + " : Since
" > 0 was arbitrary, we nally have that (s)(Ct) 6 1 t (s)(Ct);
i.e., (s)(Ct) = 0 (recall that t > 1). This proves (1.29). Next
we show that 1 2s 6 lim sup r&0 (s)(Br(x) A) (2r)s for (s)-a.a.
x 2 A: (1.31) For a given ; 2 (0; 1), we introduce the set A(; ) A,
dened by A(; ) df= x 2 A : (s) (C A) 6 (C)s for all C RN; with (C)
6 and x 2 C : Let fCngn>1 be a -cover of A(; ), such that A(; )
1[ n=1 Cn and (Cn) 6 ; and Cn A(; )6= ; 8 n > 1: So (s) A(; ) 6
1X n=1 (s) Cn A(; ) 6 1X n=1 (s) (Cn A) 6 1X n=1 (Cn)s and from
Denition 1.3.5, we see that (s) A(; ) 6 (s) A(; ) : 2005 by Taylor
& Francis Group, LLC
- 58. 48 Nonlinear Analysis Since 0 < < 1 and (s) A(; )
< +1; we have (s) A(; ) = 0: In particular, from Proposition
1.3.24, we see that (s) = 0: (1.32) A(; 1 ) Set D1 df= x 2 A : lim
sup r&0 (s)(Br(x) A) (2r)s < 1 2s : If x 2 D1, then we can
nd > 0, such that (s)(Br(x) A) (2r)s 6 1 2s 8 r 2 (0; ]: (1.33)
For any C RN, with x 2 C A and (C) 6 ; from (1.33), we have (s) (C
A) 6 (s)(C A) 6 (s) 6 (1 )(C)s: B(C)(x) A So it follows that x 2
A(; 1 ). Therefore, we have D1 1[ n=1 1 n A ; 1 1 n ; and, using
also (1.32), we have (s)(D1) = 0: Thus we infer that (1.31) is
true. For a given locally integrable function, we can establish the
Hausdor mea- sure of the set where the function is locally large.
To do this we shall need the so-called Lebesgue dierentiation
theorem or Lebesgue-Besicovitch dierentiation theorem THEOREM 1.4.6
(Lebesgue-Besicovitch Dierentiation Theorem) If f 2 L1 loc RN;RM ,
then lim r&0 1 N(Br(x)) Z Br(x) f(y) f(x) RM dN(y) = 0 for
N-a.a. x 2 RN: 2005 by Taylor & Francis Group, LLC