STRANGE FUNCTIONS IN REAL ANALYSIS
PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes
EXECUTIVE EDITORS Zuhair Nashed University of Delaware Newark, Delaware
Earl J. Taft Rutgers University New Brunswick, New Jersey
EDITORIAL BOARD M. S. Baouendi University of California, Sun Diego Jane Cronin Rutgers University Jack K. Hale Georgia Institute of Technology
Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University
S. Kobayashi University of California, Berkeley
David L. Russell Virginia Polytechnic Institute and State University
Marvin Marcus University of California, Santa Barbara
Walter Schempp Universitat Siegen
W.S. Massey Yale University
Mark Teply University of Wisconsin, Milwaukee
MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS 1. K Yano, lntegral Formulas in Rlemannian Geometry (1970) 2. S. Kobayashi, HyperbolicManlfolds and HolomorphlcMappings (1970) S. Vladlmirov, Equatlons of Mathematical Physics (A. Jeffrey, ed.; A. Littlewood, 3. trans.) (1970) 4. B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation ed.; K. Makowski, trans.) (1971) 5. L. Narlci et a/., Functlonal Analysis and Valuation Theory (1971) 6. S. S. Passman, Infinite Group Rlngs (1971) 7. L. Domhoff, Group Reprasentation Theory. Part A: Ordinary Representation Theory. Part B: Modular RepresentationTheory (1971,1972) 8. W. Boothbyand G. L. Welss, eds., Symmetric Spaces (1972) 9. Y, Matsushlma, Differentiable Manlfolds (EnT. Kobayashi, trans.) (1972) 10. L. E. Wad, Jr., Topology (1 972) 11. A. Babakhanian, Cohomologlcai Methods In Group Theory (1 972) 12. R. Gllmer, Multiplicative ideal Theory (1972) 13. J. Yeh, Stochastic Processes and the Wiener integral (1973) 14. J. Barn-Neb, lntroductlon to the Theory of Distributions (1973) 15. R. Larsen, Functlonai Analysis (1973) 16. K. Yano and S. Ishiham, Tangent and Cotangent Bundles (1973) 17. C. P~ucesl,Rlngs with Polynomial Identitles (1973) 18. R. Hennann, Geometry, Physics, and Systems (1973) 19. N, R, Wallach, Harmonlc Analysis on Homogeneous Spaces (1973) 20. J. DieudonnB, Introduction to the Theory of Formal Groups (1973) 21. 1, Vaisman, Cohomology and Differential Forms (1973) 22. B.-Y. Chen, Geometry of Submanifolds (1973) (in two .parts). (1973. 23. M. Manus. Flnlte Dimensional Multilinear Aleebra . . 1975) 24. R. Larsen, .Banach Algebras (1973) 25. R. 0.Kujala and A. L. Vifter, eds., Value DisMbution Theory: Part A; Part B: Deficit and Bezout Estimates by Wlihelm Stoll(1973) 26. K, B. Stolarsky, Algebraic Numbers and Dlophantlne Approxlmatlon (1974) 27. A. R. Magid, The Separable Galds Theory of Commutative Rings (1974) 28. 8. R. McDon~Id,Flnlte Rlngs wlth Identity (1974) 29. J. Satake, Llnear Algebra (S. Koh et ai., trans.) (1975) 30. J. S, Golan, Localizationof NoncommutativeRlngs (1975) 31. G. Klambauer, Mathematical Analysis (1975) 32. M. K Agosbn, Algebraic Topology (1976) 33. K. R. Goodearl, Ring Theory (1976) 34. L. E. MansMd, Linear Algebra with Geometric Applications (1976) 35. N. J, Pullman, Matrix Theory and Its Applications (1976) 36. B. R, McDonald, GeomeMc Algebra Over Local Rings (1976) 37. C. W. Gmtsch, Generalized Inverses of Llnear Operators (1977) 38. J. E. Kuczkowskiand J. L. Gersting, Abstract Algebra (1977) 39. C. 0.Chrlstenson and W L. Voxman, Aspects of Topdogy (1977) 40. M. Nagata, Field Theory (1977) 41. R. L. Long, Algebralc Number Theory (1977) 42. W, F. Pfehr, lntegrals and Measures (1977) 43. R. L. Wheeden and A. Zygmund, Measure and lntegral (1977) 44. J. H. Curtlss, introduction to Functions of a Complex Variable (1978) 45. K HnSecek and T. Jech, introduction to Set Theory (1978) 46. W, S. Massey, Homology and Cohomology Theory (1978) 47. M. Manus, lntroduction to Modem Algebra (1978) 48. E, C. Young, Vector and Tensor Analysis (1978) 49. S. B. Nadler, Jr., Hyperspaces of Sets (1978) 50. S. K Segal, Topics In Group Kings (1978) 51. A. C. M. van R d Non-Archimedean Functional Anal~sls(1978) . 52. L. Cowin and R.~Szczarba,Calculus in Vector Spaces (19j9) 53. C. Sadosky, lnterpolatlon of Operatora and Singular lntegrals (1979) 54. J. Cmnin, Differential Equatlons (1980) 55. C. W. Gmtsch, Elements of Applicable Functlonal Analysis (1980)
56. 1. Vaisman, Foundations of Three-DimensionalEuclidean Geometry (1980) 57. H. I, Freedan. Deterministic MathematicalModels In Po~ulatlonEcoloav -- .(1980) S. B. Chae, iebesgue Integration (1980) C. S. Rees et a/., Theory and Applications of Fourier Analysis (1981) L. Nachbin, Introduction to Functional Analysls (R. M. Aron, trans.) (1981) G. O m c h and M. Omch, Plane Algebraic Curves (1981) R. Johnsonbaugh and W, E. Pfeffenberger, Foundations of Mathematical Analysis (1981) W. L. Voxman and R. H. Goetschel, Advancad Calculus (1981) L. J. Cofwin and R. H. Szczarba, Multlvarlabie Calculus 1982) V, I, lstcftescu, Introduction to Linear Operator Theory (1 81) R. D. Jd~inen,Finite and Infinite Dimensional Linear Spaces (1981) J. K. Beem and P. E. Ehrlich, Global LorentzianGeometry (1981) D. L. Annacost, The Structure of Locally Compact Abelian Groups (1981) J. W. Brewer and M. K Smith, eds., Emmy Noether: A Tribute (1981) K H. Kim, Bodean Matrix The and Applications (1982) T. W. Wkting, The ~athemati3Theot-yof ~ h m n a t i Plane c Ornaments (1982) D. B.Gauld, Differential Topology (1982) R. L. Fabar, Foundatlons of Euclidean and Non-Euclidean Geometry (1983) M. Canneli, Statistical Theory and Random Matricas (1983) J. H. Camth et a/., The Theory of Topological Semigroups (1983) R. L. Faber, Differantlal Geometry and Relativity Theory (1983) S. Bamett, Poiynomlalsand Llnear Control Systems (1983) G. Karpibvsky, CornmutativeGroup Algebras (1983) F. Van Oystaeyen and A. Verschoren, Relative invariants of Rings (1983) I. Valsman,A Flrst Course in Differential Geometry (1984) G, W. Swan, Applicatlons of Optimal Control Theory In Biomedicine (1984) T. Pettie and J. D. Randall, Transformation Groups on Manifolds (1984) K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpanslve Mappings (1984) T. Albu and C. Nristi5sescll. Relative Finiteness In Module Theory (1984) K. Hrbacek and T. Jech. Introduction to Set Theow: Second Edition (1984) F. Van 0 staeyen and A. Verschoren, Relative lniariants of Rings (11384) . B, R. ~ c L n a l dLlnear , Algebra Over CommutativeRings (1984) M. Namba, Geometry of Projective Algebralc Curves (1984) G. F. Webb, Theory of Nonlinear Age-Dependent Populatlon Dynamics (1985) M. R. Bremner et a/., Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985 A. E. Fekete, Real Linear lgebra (1985) S. B. Chae, Hdomomhv and Calculus In Normed S~aces(1985) 93. A. J. Jeni, introduction h Integral Equations with Appllcatidns (1985) 94. G. Karpiovsky, Projective Representationsof Finlte Groups (1985) 95. L. Narlci and E. Beckenstein. To~oloalcalVector S~aces,119851 . 96. J. Weeks, The Shape of ~ p a c a(1988) 97. P. R. Gdbik and K. 0. Kortanek, Extremal Methods of 0 rations Research (1985) 98. J,-A. Chao and W. A, Woyczynski, eda., Probability # k r y and H a m n i c Analysis 11986) 99. h~D. ~ m w et n a/.. Abstract Algebra (1986) 100. J. H. Cemth eta/.,The Theory of Topological Semlgroups, Volume 2 (1986) 101. R. S. Domn and K A. Bern, CharacterizationsofC*-Algebras (1986) 102. M. W. Jeter, Mathematical Programming (1986) 103. M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications (1986) 104. A. Verschoren, Relative lnvarlants of Sheaves (1987) 105. R. A. Usmani, Applied Linear Algebra (1987) 106. P. Blass and J. Lang, Zariski Surfacas and Differantlai Equations in Characteristic p > 0 (1987) 107. J. A. Reneke et ab, Structured Hereditery Systems (1987) 108. H, Busemann and B. B. Phadke, Spaces with Distinguished Geodesics (1987) 109. R. Ha&, lnvertlbillty and Singularity for Bounded Linear Operators (1988) 110. G. S. Ladde et a/., Oscillation Theory of Differential Equations with Deviating Arguments ( I 987) I II. L. Dudkin et ab, Iterative AggregationTheory (1987) 112. T.' Okubo, Differential Geometry (1987) 7
Q
A
D. L. Stancl and M. L. Stanel, Real Analysis with Point-Set Topology (1987) T. C. Gad, introduction to Stochastic DlfferentlalEquations (1988) S. S. Abhyankar, Enumerative Combinatorlcs of Young Tableaux (1988) H. Strade and R. Famsteiner, Modular Lie Algebras and Their Representations (1988) J. A. Huckaba, Commutative Rings with Zero Divisors (1988) W. D. Wallis, Combinatorlal Desi ns (1988) W, W@HWTopological Fields (7988) G, Karpilovsk Field Theory (1988) S CaenepeerandF. Van Oystaeyen, Brauer Groups and the Cohomoiogy of Graded Rinas . ....-- (IQt3Q) W. Kozlowskl, Modular Function Spaces (1988) E, Lowen-Cdebunders,Function Classes of Cauchy Continuous Maps (1989) M. Pawl, Fundamentals of Pattern Recognition (1989) V. Lakshmikanthamet a/.. Stabiiltv Analvsis of Nonlinear Svstems 119891 R. Sjwmmakrishnan, he ~iassiiai T h k y of Arithmetic ~inctions(1989) N. A. Watson. Parabolic Eauations on an Infinite Strip (1989) K. J. Hastings, Introductlon'to the Mathematics of Opektions Research (1989) B. Flne, Algebraic Theory of the Blanchi Groups (1989) D. N. Djkranjanet a/,, Topoiogicai Groups (1989) J. C. Morgan 11, Pdnt Set Theory (1990) P. Biler and A. Wtkowski, Problems in Mathematical Analysis (1990) H. J. Sussmann, Nonlinear Controllability and Optimal Control (1990) J.-P. Florenset a/., Elements of Bayesian Statistics (1990) N. Shell, Topologicai Fieids and Near Valuations (1990) B. F. Doolin and C. F. Martin, introduction to Differential Geometry for Engineers
.----,
FIm\ ,11.---,
151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171.
S. S. Hdland, Jr., Applied Analysis by the Hilbert Space Method (1990) J. Oknlnski, Semigroup Algebras (1990) K Zhu, Operator Theory in Function Spaces (1990) G. 8. Price, An introduction to Multicomplex Spaces and Functions (1991) R. B. Darst, introduction to Linear Programming (1991) P. L. Sachdev, NonlinearOrdina Dlfferentlal Equations and Their Appilcatlons(1991) Z Husain, Otthogonal ~chauder%ases(1991) J. Foren, Fundamentalsof Real Analysis (1991) W, C. Bmwn, Matrices and Vector Spaces (1991) M, M. Rao and 2. D. Ren, Theory of Oriicz Spaces (1991) J. S. Golan and T. Head, Modules and the Structures of Rings (1991) C. Small, Arithmetic of Flnite Fields (1991) K Yang, Complex Algebraic Geometry (1991) D. G. Hoffman et 81.. Codina Theow 11991) M. 0. GonzBk, ~lasslcalb m p i e i ~nalyils(1992) M. 0. GonzBlez, Complex Analysis (1992) L. W. Baggeft, Functional Analjkis (I992 j M. Sniedovich, Dynamic Programming (1992) R. P. Agawal, Difference Equations and inequalities (1992) C. Brezlnskl, Biorthogonaiity and its Appiicationsto Numerical Analysis (1992) C. Swartz, An Introduction to Functional Analysis (1992) S. B. Nadler, Jr., Continuum Theo (1992) M. A. ACGwaiz. ~heorv of ~istribugns(1992) E. Peny, Geometry: hiomatic eve lop in en ti with Problem Solving (1992) E. Castill0 and M. R. Rulz-Cobo, Functional Equations and Modelling in Sdence and Engineerlng (1992) A. J. Jerrl, Integral and Discrete Transforms with Applications and Error Analysis (1992) A. Charlleret el., Tensors and the Clifford Algebra (1992) P, Biler and T. Nadzleja, Problems and Exam ies in Differential Equations (1992) E. Hansen. Global OptimizatlonUsing lntervaP~nalysls (1992) S. Guenw-Delabtfdre,Classical Sequences In Banach Spaces (1992) Y. C. Wong, Introductory Theory of Topdogical Vector Spaces (1992) S. H. Kulkamiand 8. V. Limaye, Real Function Algebras (1992) W. C. Bmwn, Matrices Over Commutative Rings (1993) J. Loustau and M. Dillon, Linear Geometry with Computer Graphics (1993) W. V. Pettyshyn, Approximatlon-Soivabliity of Nonlinear Functional and Differential Equations (1993)
172. 173. 174. 175.
E. C. Young, Vector and Tensor Analysis: Second Edition (1993)
T. A. Bick. Eiementarv Boundarv Value Problems (1993) M. ~ave1,'~undameniais of p a t h Recognition: ~ k n Editlon d (1993) S. A. Albeverio et a/., Noncommutatlve Distributions (1993) . . W Fulks, Complex Variables (1993) M. M, Rao, Conditional Measures and Appiications (1993) A. Janicki and A. Wemn, Slmuiatlon and Chaotic Behavior of a-Stable Stochastic Processes (1994) P. Neitfaanmdkiand D. Tlbs, Optlmal Control of Nonlinear Parabolic Systems (1994) J. Cmnin, Differential Equations: introduction and Qualitative Theory, Second Edition (1994) S. Heikkild and V. Lakshmikantham, Monotone iterative Techniques for Discontinuous Nonlinear Differential Equatlons (1994) Mao, Exponential Stabili of Stochastic Differential Equatlons (1994) B. S Thomwn, Symmetric%ro erUes of Real Functions (1994) J. E Rubio, Optimization and dnstandad Analysis (1994) J. L. Buesoet a/., Compatibiiity, Stability, and Sheaves (1995) A. N. Micheland K Wang, Qualitative Theory of Dynamicai Systems (1995) M. R. Damel, Theory of Lattice-OrderedGroups (1995) Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational inequalities and Applications (1995) L. J. Coiwinand R. H. Szczan5e, Calculus in Vector Spaces: Second Edition (1995) L. H. Enbe eta/,, Oscillation Theory for Functional Differential Equations (1995) S. Agaian et a/., Binary Polynomial Transforms and Nonlinear Digital Filters (1995) M. I.Gil: Norm Estimations for Operatlon-Valued Functions and Appiicatlons (1995) P. A. Grillet, Semigroups: An lntroduction to the Structure Theory (1995) S. Kichenassamy,Nonlinear Wave Equatlons (1996) V. F. Kmtov, Global Methods in Optimal Control Theory (1996) K. I.Beidar et a/., Rings wlth Generailzed identities (1996) K I. Amautov et a/., lntroduction to the Theory of Topoioglcai Rings and Modules 119961
199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209. 210. 211. 212. 213. 214.
G. ~iehsrna.Linear and lnterrer Pmrammina (1996). R. Lasser, introduction to Fotrler S d e s (19a)' K Slma, Al~orlthmsfor Linear-Quadratic Optimization (1996) . . D. Redmond, Number Theory (1996) J. K. Beem eta/., Global LorentzlanGeometry: Second Edition (1996) M Fontanaet a/., PrClfer Domains (1997) H. Tanabe, Functional Analytic Methods for Partial Differential Equations (1997) C. Q. Zhang, integer Flours and Cycle Covers of Graphs (1997) E. Spiegeland C. J. O'Donnell, inddenca Algebras (1997) B. Jakubczyk and W. Respondek, Geometry of Feedback and 0 timai Control (1998) T. W Haynes et a/,, Fundamentals of Domination In Graphs (1d8) T. W. Haynes et a/., Domination in Graphs: Advanced Toplcs (1998) L. A. D'Alotto et a/,, A Unified Signal Algebra Approach to Two-Dimensional Parallel Digital Signal Processing (1998) F. Halter-Koch, ideal Systems (1998) N. K. Govilet al., Approximation Theory (1998) R. Cms, Multivalued Linear Operators (1998) A. A. Martynyuk, Stabiiity by Liapunov's Matrix Function Method with Applications
QQAI ,l l.""-,
215. A. Favlni and A. Yagi, Degenerate Differential Equatlons in Banach Spaces (1999) 216. A. lllanes and S. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances 1lQQQ\ ,----I
217. G. Kato and D. Stnr , Fundamentals of Algebraic Microlocal Analysis (1999) 218. G. X-Z. Yuan, K K heory ~ ?and Applications in Nonlinear Analysis (1999) 219. D. Motreanu and N. H. Pavel, Tangency, Flow lnvariance for Differential Equations, and Optimization Problems (1999) 220. K. Hrbacek and T. Jech, lntroduction to Set Theory, Third Edition (1999) 221. G. E. Kolosov, Optimai Design of Control S stems (1999) 222. N. L Johnson, Subpiane Covered Nets (200g) 223. B. Fine and G. Rosenbqer, Algebraic Generalizationsof Discrete Groups (1999) 224. M. Vdth, Voiterra and integral Equations of Vector Functions (2000) 225. S. S. Miller and P. T. Mocanu, Differential Subordinations (2000)
226.
R. Li et a/., Generalized Difference Methods for Differential Equations: Numerical
Analysis of Finlte Volume Methods (2000) 227. H. Li and F. Van Oysfaeyen,A Prlmer of Algebraic Geometry (2000) 228. R. P. Agawal, Difference Equatlons and Inequaiitles: Theory, Methods, and Appiicatlons, Second Edition (2000) 229. A. B. Kharazishvili, Strange Functions in Real Analysis (2000) Additional Volumes in Preparation
STRANGE FUNCTIONS IN REAL ANALYSIS A. B.Kharazishvili Tbllisi State University Tbilisi, Republic of Georgia
MARCLL
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Preface
At the present time, many strange (or singular) objects in various fields of mathematics are known and no working mathematician is greatly surprised if he meets some objects of this type during his investigations. In connection with strange (singular) objects, the classical mathematical analysis must be noticed especially. It is sufficient to recall here the wellknown examples of nowhere differentiable real-valued functions; examples of Lebesgue measurable real-valued functions nonintegrable on any nonempty open subinterval of the real line; examples of Lebesgue integrable realvalued functions with everywhere divergent Fourier series, and others. There is a very powerful technique in modern mathematics by means of' which we can obtain various kinds of strange objects. This is the socalled category method based on the classical Baire theorem from general topology. Obviously, this theorem plays one of the most important roles in mathematical analysis and its applications. Let us recall that, according to the Baire theorem, in any complete metric space E (or, in any locally compact topological space E ) the complement of a first category subset of E is everywhere dense in E, and it often turns out that this complement consists precisely of strange (in a certain sense) elements. Many interesting applications of the category method are presented in the excellent textbook by Oxtoby [117] in which the deep analogy between measure and category is thoroughly discussed as well. In this connection, the monograph by Morgan [110]must also be pointed out where an abstract concept generalizing the notions of measure and category is introduced and investigated in detail. Unfortunately, the category method does not always work and we somet i m eneed s an essentially different approach to questions concerning the existence of singular objects. This book is devoted to some strange functions in real analysis and their applications. Those functions can be met in various studies in analysis and play an essential role there, especially as counterexamples to numerous statements which seem t o be very natural but, finally, fail to be true in certain situations (see, e.g., [49]).Another important role of strange functions, with respect to given concepts of analysis, is to show that those concepts
iv
P REF A C E
are, in some sense, not satisfactory and hence have to be revised, generalized or extended in an appropriate direction. In this context, we may say that strange functions stimulate the development of analysis. The book deals with a number of important examples and constructions of strange functions (primarily, we consider functions acting from the real line into itself). Notice that many such functions can be obtained by using the category method (for instance, a real-valued continuous function defined on the closed unit interval of the real line, which does not possess a finite derivative a t each point of this interval). But, as mentioned above, there are some situations where the classical category method cannot be applied, and thus, in such a case, we have t o appeal to the corresponding individual construction. We begin with functions which can be constructed within the theory ZF & DC where ZF denotes the Zermelo-Fraenkel set theory without the Axiom of Choice and DC denotes a certain weak form of this axiom: the so-called Axiom of Dependent Choices which is enough for most domains of classical mathematics. Among strange functions whose existence can be established in ZF & DC the following ones are of primary interest: Can. tor and Peano type functions, everywhere differentiable nowhere monotone functions, Jarnik's nowhere approximately differentiable functions. Then we examine various functions whose constructions need essentially noneffective methods, i.e. they need an uncountable form of the Axiom of Choice: functions nonmeasurable in the Lebesgue sense, functions without the Baire property, functions associated with a Hamel basis of the real line, Sierpiriski-Zygmund functions which are discontinuous on each subset of the real line of the cardinality continuum, etc. Finally, we consider a number of examples of functions whose existence cannot be established without additional set-theoretical axioms. However, it is demonstrated in the book that the existence of such functions follows from (or is equivalent to) certain widely known set-theoretical hypotheses (e,g, the Continuum Hypothesis). Among other topics presented in this book, closely connected with strange functions in real analysis, we wish to point out the following ones: Egorov's and Mazurkiewicz's theorems on uniform convergence of measurable functions, some relationships between the classical Sierpiriski partition of the Euclidean plane and Fubini type theorems, sup-measurable and weakly sup-measurable functions with their applications in the theory of ordinary differential equations. In the final chapter of our book, we consider the family of all nondifferentiable functions from the points of view of category and measure. We present one general approach illuminating the basic reasons which necessarily imply that the above-mentioned family of functions has t o be large in
PREFACE
V
the sense of category or measure. Notice that, in connection with nondifferentiable functions, a short scheme for constructing the classical Wiener measure is discussed in this chapter, too, and some simple but useful statements from the general theory of stochastic processes are demonstrated. This book is based on the course of lectures given by the author at Institute of Applied Mathematics of Tbilisi State University in the academic year 1997 1998, entitled:
-
Some Pathological Functions in Real Analysis. These lectures (their role is played by the corresponding chapters of the book) are, in fact, mutually independent from the logical point of view but are strictly related from the point of view of the topics discussed and the methods applied (such as purely set-theoretical arguments and constructions, measure-theoretical methods, the Baire category method, and so on). The material presented in the book is essentially self-contained and, consequently, is accessible to a wide audience of mathematicians (including graduate and postgraduate students). For the reader's convenience, Chapter 0 plays the role of introduction to the subject. Here some preliminary notions and facts are given that are useful in our further considerations. The reader can ignore this auxiliary chapter, returning to it if the need arises. In this connection, the standard graduate-level textbooks and monographs (for instance, [53], [66], [91], [117], [127]) should be pointed out containing all preliminary notions and facts from set theory, general topology and real analysis. We begin with basic set-theoretical concepts such as: binary relations of special type (namely, equivalence relations, orderings, functional graphs), ordinal numbers, cardinal numbers, the Axiom of Choice and the Zorn lemma, some weak forms of the Axiom of Choice (especially, the countable form of AC and the Axiom of Dependent Choices), the Continuum Hypothesis, the Generalized Continuum Hypothesis, and Martin's Axiom as a set-theoretical assertion which is essentially weaker than the Continuum Hypothesis but rather helpful in various constructions of set theory, topology, measure theory and real analysis. Then we briefly present some basic concepts of general topology and classical descriptive set theory, such as: the notion of a first category set in a topological space, the Baire property (the Baire property in the restricted sense) of subsets of a topological space, the notion of a Polish space, Borel sets in a topological space, analytic (Suslin) subsets of a topological space, and the projective hierarchy of Luzin, which takes the Borel and analytic sets as the first two steps of this hierarchy. It is also stressed that Borel and
vi
PREFACE
analytic sets have a nice descriptive structure but this feature fails to be true for general projective sets (since, in certain models of set theory, there exist projective subsets of the real line which are not Lebesgue measurable and do not have the Baire property). The final part of Chapter 0 is devoted to some classical facts and statements from real analysis. Namely, we recall here the notion of a real-valued lower (upper) semicontinuous function and demonstrate basic properties of such functions, formulate and prove the fundamental Vitali covering theorem, introduce the notion of a density point for a Lebesgue measurable set, and present the Lebesgue theorem on density points as a consequence of the above-mentioned Vitali theorem. In addition, we give here a proof of the existence of a nowhere differentiable real-valued continuous function, starting with the well-known Kuratowski lemma on closed projections. Let us emphasize once more that the question of the existence of real-valued continuous nondifferentiable functions, with respect to various concepts of generalized derivative, is one of the central questions in this book. We develop this topic gradually and, as mentioned earlier, investigate the question from different points of view; however, the main focus is on its purely logical and set-theoretical aspects.
Contents
. . . . . . . . . . . . . . . . . . . . . . . . . iii Introduction: basic concepts . . . . . . . . . . . . . . 1 Cantor and Peano type functions . . . . . . . . . . . . 33 Singular monotone functions . . . . . . . . . . . . . . 55 Preface..
Everywhere differentiable nowhere monotone functions. ..
........... ............ Nowhere approximately differentiable functions . . .
69 83
I
Blumberg's theorem and Sierpiliski-Zygmund function.
95
Lebesgue nonmeasurable functions and functions without the Baire property
113
.... ....... .. .... . ........
... .... .. ...... Hamel basis and Cauchy functional equation . . . . ..
159
.................
181
Luzin sets, Sierpinski sets and their applications Egorov type theorems.
137
Sierpinski's partition of the Euclidean plane
. . . . . 197
Sup-measurable and weakly sup-measurable functions.
217
... .......... .. .. .... . .. .
Ordinary differential equations with bad right-hand sides
. . . . . . . . . . . . . . . . . . . . . . 237
viii
.
13
CONTENTS
Nondifferentiable functions from the point of view of category and measure
. . . . . . . . . . . . . . . . 251 Bibliography . . . . . . . . . . . . . . . . . . . . . . . 279 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
0. Introduction: basic concepts
In this chapter we fix the notation and present some elementary facts from set theory, general topology and the theory of real functions. We shall systematically utilize these facts in our further considerations. The symbol ZF denotes the Zermelo-Fraenkel set theory which is one of the most important formal systems of axioms for the whole of modern mathematics (in this connection, see [88] and [91]; cf. also [13]). The basic notions of the Zermelo-Fraenkel system are sets and the membership between them. Of course, the system ZF consists of several relation axioms which formalize various properties of sets in terms of the relation We do not present here a list of these axioms and, actually, we shall work in the so-called "naive set theory". The symbol ZFC denotes the Zermelo-Fraenkel theory with the Axiom of Choice. In other words, ZFC is the following theory:
where AC denotes, as usual, the Axiom of Choice. At the present time, it is widely known that the theory ZFC is a basis of modern mathematics, i.e. almost all fields of mathematics can be developed by starting with ZFC.The Axiom of Choice is a very powerful set-theoretical assertion which implies many extraordinary and interesting consequences. Sometimes, in order t o get a required result, we do not need the whole power of the Axiom of Choice. In such cases, it is sufficient to apply various weak forms of AC.Some of these forms are discussed below. If x and X are any two sets, then the relation x X means that belongs to X. In this situation, we also say that x is an element of X. One of the axioms of set theory implies that any set y is an element of some set Y (certainly, depending on y). Thus we see that the notion of an element is equivalent to the notion of a set. The relation X Y means that a set X is a subset of a set Y . The relation X c Y means that a set X is a proper subset of a set Y . If R(x) is a relation depending on an element x (or, in other words,
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R(x) is a property of an element x), then the symbol
denotes the set (the family, the class) of all those elements x for which the relation R(x) holds. In our further considerations we always suppose that R(x) is such that the corresponding set {x : R(x)} does exist. For example, a certain axiom of ZF states that there always exists a set of the type {x : x X S(x)} where X is an arbitrarily given set and this case we write X : instead of {x : x X then we write instead of {x : The symbol
is an arbitrary relation. In
Also, if we have two relations
and
denotes, as usual, the empty set,
If X is any set, then the symbol denotes the family of all subsets we have of X , = {Y : Y The set is also called the power set of a given set X . If x and y are any two elements, then the set
is called the ordered pair (or, simply, the pair) consisting of x and viously, we have the implication ((x, y) =
(x =
y=
for all elements x , x', y'. Let X and Y be any two sets. Then, as usual, X U Y denotes the union of X and Y; X fl Y denotes the intersection of X and Y; X \ Y denotes the difference of X and Y; X A Y denotes the symmetric difference of X and Y, i.e. XAY = ( X \ Y ) u ( Y \X).
Ob-
INTRODUCTION
We also put X x Y = {(x,y)
:
XEX, y EY).
The set X x Y is called the Cartesian product of the given sets X and Y. In a similar way, by recursion, we can define the Cartesian product
of a finite family {XI, X2,... , X,) of arbitrary sets. If X is a set, then the symbol card(X) denotes the cardinality of X . Sometimes, card(X) is also called the cardinal number of X . w is the first infinite cardinal (ordinal) number. In fact, w is the cardinality of the set N = {0, 1, 2, , n, ,,.) ..#
of all natural numbers. Sometimes, it is convenient to identify the sets w and N. A set X is finite if card(X) < w . A set X is infinite if card(X) 1 w . A set X is (at most) countable if card(X) 5 w . Finally, a set X is uncountable if card(X) > w . For an arbitrary set E, we put: = the family of all finite subsets of E; [E]sW= the family of all countable subsets of E . w l is the first uncountable cardinal (ordinal) number. Notice that w l is sometimes identified with the set of all countable ordinal numbers (countable ordinals). Various ordinal numbers (ordinals) are denoted by a, p, y, E, C, ... . Let a be an ordinal number. We say that a is a limit ordinal if a = sup{@ :
p < a).
The cofinality of a limit ordinal a is the smallest ordinal 5 such that there exists a family {aS : C < E) of ordinals satisfying the relations (VC < E ) ( q < a ) ,
a = sup{ac :
C <El.
The cofinality of a limit ordinal a is denoted by the symbol cf (a). Clearly, we have the inequality cf ( a ) a for all limit ordinal numbers
<
a.
A limit ordinal number a is called a regular ordinal if cf(a) = a. A limit ordinal number a is called a singular ordinal if cf ( a ) < a. Starting with the definitions of regular and singular limit ordinals, we can define, in the usual way, regular infinite cardinals and singular infinite
4
CHAPTER 0
cardinals. For example, w and w l are regular ordinals (cardinals) and w, is a singular ordinal (cardinal). In many considerations it is convenient to identify every ordinal CY with the set of all those ordinals which are strictly less than a. Such an approach to the theory of ordinal numbers is due to von Neumann (see [88], Chapter 1). It is also convenient to identify every cardinal K with the smallest ordinal number a such that card(a) = n. If n is an arbitrary infinite cardinal number, then the symbol K+ denotes the smallest cardinal among all those cardinals which are strictly greater than n. For example, we have
The symbol Z denotes the set of all integers. The symbol Q denotes the set of all rational numbers. The symbol R denotes the set of all real numbers. If the set R is equipped with its standard structures (order structure, algebraic structure, topological structure), then R is usually called the real line. The symbol c denotes the cardinality of the continuum, i.e, we have
The Continuum Hypothesis ( C H ) is the assertion
The Generalized Continuum Hypothesis ( G C H ) asserts that
for all ordinals a. At the present time, it is well known that the theory Z F C is consistent if and only if the theories Z F C & (the Continuum Hypothesis), Z F C & (the negation of the Continuum Hypothesis) are consistent (see, e.g., [88], Chapter 7). Moreover, it is also well known that the theory Z F C is consistent if and only if the theories ZFC & (c is a regular cardinal), Z F C & (c is a singular cardinal)
INTRODUCTION
5
are consistent. More precisely, it was established that, for an infinite cardinal number w, satisfying the relation cf (w,) > w , there exists a model of ZFC in which we have the equality c = w,. Actually, if we start with an arbitrary countable transitive model for ZFC (strictly speaking, for a relevant fragment of ZFC) satisfying the Generalized Continuum Hypothesis, then the above-mentioned equality is true in a certain Cohen model for ZFC extending the original model (for details, see [88], Chapter 7). The Generalized Continuum Hypothesis holds in a special model of set theory, first constructed by Godel. This model is called the' Constructible Universe of Godel and usually denoted by L. Various facts and statements concerning L are discussed in [88], Chapter 6 (see also [55] and [56]). It is reasonable to note here that, in L, some naturally defined subsets of the real line are bad from the point of view of Lebesgue measure and Baire property (i.e. they are not measurable in the Lebesgue sense and do not have the Baire property). Let n be a fixed natural number. The symbol Rn denotes, as usual, the n-dimensional Euclidean space. If n = 0, then Rn is the one-element set consisting of zero only. If n > 0, then it is sometimes convenient to consider R n as a vector space V over the field Q of all rational numbers. According to a fundamental assertion of the theory of vector spaces (over arbitrary fields), there exists a basis in the space V (see, e.g., [31] where much more general statements are discussed for universal algebras). This basis is usually called a Hamel basis of V . Obviously, the cardinality of any Hamel basis of V is equal to the cardinality of the continuum. Notice also that the existence of a Hamel basis of V cannot be established without the aid of uncountable f o r m of the Axiom of Choice because the existence of such a basis immediately implies the existence of a subset of the real line R, nonmeasurable with respect to the standard Lebesgue measure given on R. Let X and Y be any two sets. A binary relation between X and Y is an arbitrary subset G of the Cartesian product of X and Y , i.e.
In particular, if we have X = Y , then we say that G is a binary relation on the basic set X. For any binary relation G C X x Y , we put
~7'2(G)= {y : (3x)((x, y) E G)). It is clear that G C prl(G) x prz(G).
6
CHAPT ER
0
The Axiom of Dependent Choices is the following set-theoretical statement: If G i s a binary relation on a nonempty set X and, for each element x E X , there exists an element y E X such that (x, y) E G, then ihere exists a sequence (xo, X I , ..., xn, ...) of elements of the set X , such that
The Axiom of Dependent Choices is usually denoted by D C . Actually, the statement D C is a weak form of the Axiom of Choice. This form is completely sufficient for most fields of classical mathematics: geometry of a finite-dimensional Euclidean space, mathematical analysis of the real line, Lebesgue measure theory, etc. We shall deal with the axiom D C many times in our further considerations and discuss some interesting applications of this axiom. It was established by Blair that, in the theory ZF, the next two assertions are equivalent: a) the Axiom of Dependent Choices; b) no nonempty complete metric space is of the first category (the classical theorem of Baire). Exercise 1. Prove the logical equivalence of assertions a) and b) in the theory Z F . Note that implication a) =+ b) is widely known in analysis. In order to establish the converse implication, equip a nonempty set X with the discrete topology and consider the complete metric space X W . Further, by starting with a given binary relation G on X satisfying
define a certain countable family of dense open subsets of X W and obtain with the aid of this family the desired sequence of elements from X . Let X be an arbitrary set. A binary relation G C X x X is called an equivalence relation on the set X if the following three conditions hold: 1) (x, x) E G for all elements x E X ; 2) (2, y) E G and (y, z) E G imply (8, z) E G; 3) (x,y) E G implies (y,x) E G. If G is an equivalence relation on X , then the pair ( X , G) is called a set equipped with an equivalence relation. In this case, the set X is also called the basic set for the given equivalence relation G.
7
INTRODUCTION
Obviously, if G is an equivalence relation on X , then we have a partition of X canonically associated with G. This partition consists of the sets
where G(x) denotes the section of G corresponding to an element x E X ; in other words, G(x) = {y : (a:,!I) E G I . Conversely, every partition of a set X canonically defines an equivalence relation on X . Let X be an arbitrary set and let G be a binary relation on X . We say that G is a partial order on X if the following three conditions hold: (1) (x, x) E G for each element x E X ; (2) (x, y) E G and (y, z) E G imply (x, z) E G; (3) (x,y) E G and (y,x) E G imply x = y. Suppose that G is a partial order on a set X . As usual, we write
iff
(x,y) E G.
The pair (X, 5 ) is called a set equipped with a partial order (or, simply, a partially ordered set). The set X is called the baaic set for the given partial order 5 . Let ( X , < ) be a partially ordered set. Let x and y be any two elements of the basic set X . We say that x and y are comparable (with respect to -<) if x s y v ysx. According to this definition, we say that x and y are incomparable if
Further, we say that elements x and y of the set X are consistent (in X ) if there exists an element z of X such that z 5 x and z y. Now, it is clear that elements x and y of the set X are inconsistent (in X ) if there does not exist an element z of X having the above property. Obviously, if x and y are inconsistent, then they are incomparable. The converse assertion is not true in general. A subset Y of the set X is called a chain in X (or a subchain of X ) if any two elements of Y are comparable with respect to 5 . A subset Y of the set X is called an antichain in X if any two distinct elements of Y are incomparable with respect to 5 . In this case, Y is also called a free subset of X with respect to 5.
<
8
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A subset Y of the set X is called consistent if, for any two elements yl and ya of Y , there exists an element y of Y such that y yl and y y2. From this definition it immediately follows, by induction, that if a nonempty set Y E X is consistent, then, for every finite set { y l , y2, ... , y n ) S Y, there exists an element y of Y satisfying the relations
<
<
A subset Y of the set X is called totally inconsistent if any two distinct elements of Y are inconsistent in X . The following definition is very important for modern set theory and its various applications to general topology and measure theory (see, e.g., [881). Let (X, <) be a partially ordered set. We say. that (X, <) satisfies the countable chain condition (or the Suslin condition) if every totally inconsistent subset of X is at most countable. Let ( X , < ) be a partially ordered set and let Y be a subset of X . We say that Y is coinitial in X if, for each element x of X , there exists an element y of Y such that y x. Martin's Axiom is the following set-theoretical statement:
<
If (X, 5 ) is a partially ordered set satisfying the countable chain condition and F is a family of coinitial subsets of X I such that card(F) < c, then there exists a consistent subset Y of X which intersects each set from the family F, i,e. (Vz E F)(Y nz # 0). Martin's Axiom is usually denoted by M A . One can easily show that the Continuum Hypothesis implies Martin's Axiom. Exercise 2. Prove in ZFC the implication CH s-MA.
On the other hand, it was established (by Martin and Solovay) that if the theory ZFC is consistent, then the theory ZFC & MA & (the negation o f the Continuum Hypothesis)
is consistent, too (see e.g. [88], Chapter 8). Moreover, it was established that Martin's Axiom is a set-theoretical statement much weaker than the Continuum Hypothesis because it does not bound from above the size of
INTRODUCTION
9
the continuum. At the present time, many applications of Martin's Axiom to the theory of infinite groups, to general topology and to measure theory are known (see e.g. [55], [56], [88]). Some nontrivial applications of MA will be discussed below. Note, in addition, that M A can also be formulated in purely topological terms (see [55], [56], [88]), Let (X, 5 ) be again a partially ordered set. We say that (X, 5 ) is a linearly ordered set if any two elements of X are comparable with respect t o 5 . A linearly ordered set is also called a chain. We say that an element x of a partially ordered set (X, 5 ) is maximal (in X ) if, for each element y of X , we have the implication
In a similar way one can define a minimal element of a partially ordered set (X, 5 ) . The next set-theoretical statement is a well-known equivalent of the Axiom of Choice, formulated in terms of partially ordered sets. This statement is usually called the Zorn Lemma. Let (X, 5 ) be a partially ordered set such that each subchain of X is bounded from above. Then there exists at least one maximal element in X . Moreover, if x is an arbitrary element of X , then there exists at least one maximal element y in X satisfying the inequality x 5 y.
Sometimes, it is convenient to apply the Zorn Lemma instead of the Axiom of Choice. Let (X, 5 ) be a partially ordered set and let x E X . We say that x is a smallest (first, least) element of X if x 5 y for all elements y E X . In a similar way we can define a largest (last, greatest) element of X . It is easy to check that a smallest (respectively, largest) element of X is unique. Moreover, the smallest (respectively, largest) element of the set X is a unique minimal (respectively, unique maximal) element of X . We say that a partially ordered set (X, 5 ) is well ordered if any nonempty subset of X , equipped with the induced order, has a smallest element. Obviously, every well ordered set is linearly ordered. Well ordered sets are very important in the class of all partially ordered sets because one can directly apply the principle of transfinite induction and the method of transfinite recursion to well ordered sets. It is known that, for every well ordered set (X, 5 ) , there exists a (unique) ordinal number a such that (X, 5 ) and a are isomorphic as partially ordered sets. Thus, without loss of generality, well ordered sets can be identified with ordinal numbers.
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Let (X, 5 ) be an arbitrary linearly ordered set. Then it is not difficult to prove (by using the Zorn Lemma) that there exists a subset Y of X satisfying the following two relations: 1) Y is a well ordered set with respect to the induced order; 2) Y is a cofinal subset of X , i.e, for each element x of X , there is an element y of Y such that x _< y, This fact immediately implies that the Zorn Lemma can be formulated in the weaker version: Let (X, 5 ) be a partially ordered set such that every well ordered subset of X is bounded from above. Then there exists a maximal element in X . Moreover, if x is an arbitrary element of X , then there exists a maximal element y in X such that x 5 y. Now, we shall consider some simple facts concerning the fundamental notion of a function (or a mapping). Let X and Y be any two sets. Suppose that G is a binary relation between X and Y, i.e. G E X x Y. We say that G is a functional graph if the implication
holds for all elements x , y, y'. It is easy to see that a binary relation G S: X x Y is a functional graph if and only if
We say that a triple
s = (G,X,Y) is a partial function (or a partial mapping) acting from X into Y if G is a functional graph and G C X x Y. In this case, we also say that the set G is the graph of a partial function g. Furthermore, we say that a triple g = (G, X , Y) is a function (or a mapping) acting from X into Y if g is a partial function acting from X into Y and X = prl(G). In this case, we also write
If x is an arbitrary element of X , then the symbol g(x) denotes the unique element y of Y for which (x, y) E G. The element g(x) is called the value of g at x. Hence we can write
11
INTRODUCTION
We may use a similar notation for any partial function g = (G, X , Y), too. For example, it is sometimes rather convenient to write g : X -+ Y for this partial function. But we wish to emphasize that the symbol g(x) can be applied only in the case when x E prl(G). Let g = (G, X, Y) be again a partial mapping acting from X into Y. If A is a subset of the set X , then we put
The set g(A) is usually called the image of A with respect to g. Obviously, one can introduce, by the same definition, the set g(A) for an arbitrary set A. If B is a subset of the set Y , then we put g-l(~= ) {x : x E prl (G), g(x) E B ) . The set g - l ( ~ )is usually called the preimage of B with respect to g. Clearly, one can introduce, by the same definition, the set g-l(B) for any set B . If A is a subset of X , then the symbol glA denotes the restriction of g to this subset, i.e. we put
Evidently, the same definition can be applied to an arbitrary set A. We say that a partial function g is an extension of a partial function f i f f is a restriction of g. The set prl(G) is called the domain of g. It is denoted by dom(g). The set prz(G) is called the range of g. It is denoted by ran(g). Obviously, we have the equality ran(g) = g(dom(g)). We say that a partial function g = ( G , X , Y) is an injective partial function (or, simply, an injection) if the implication
holds for all elements x, XI from the domain of g. If g = (G, X , Y) is injective, then we have a partial function
whose graph is the set G-I
= {(y, x)
: (x, y) E G).
This partial function is called the partial function inverse to g.
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We say that a partial function g = (G, X , Y) is a surjective partial function (or, simply, a surjection) if the equality ran(g) = Y is fulfilled. Finally, we say that a function g = (G, X , Y) is a bijective function (or, simply, a bijection) if g is an injection and a surjection simultaneously. In this case, we also say that g is a one-to-one correspondence between the sets X and Y. A transformation of a set X is an arbitrary bijection acting from X onto X . The set of all transformations of a set X becomes a group with respect to the natural operation of composition of transformations. This group is called the symmetric group of X and denoted by the symbol Sym(X). The group Sym(X) is universal in the following sense: if (I?, is an abstract group such that card(I') 5 card(X), then there exists a subgroup of Sym(X) isomorphic to I?. a)
Suppose that I is a set and g is a function with dom(g) = I. Then we say that ( g ( i ) ) i ~ r (or {g(i) : a E I ) ) is a family of elements indexed by I. In this case, we also say that I is the set of indices of the family mentioned above. Moreover, suppose that E is a fixed set and, for each index i E I, the element g(i) coincides with a subset Fi of E. Then we say that
is an indexed family of subsets of E. Actually, in such a case, we have a certain mapping
F : I -+ P(E) where P ( E ) denotes the family of all subsets of E. A mapping of this type is usually called a set-valued mapping (or a multi-valued mapping). As we know, the graph of F is the set
{(i,F(i)) : i E I) C I x P ( E ) . But if we treat F as a set-valued mapping, then it is sometimes useful to consider another notion of the graph of F. Namely, the graph of a set-valued mapping F is the following set:
Obviously, the concept of a set-valued mapping is more general (in some sense) than the concept of an ordinary mapping. Indeed, every ordinary mapping JP : X - + Y
.
INTRODUCTION
can be regarded as a set-valued mapping
of a special type; namely, for any element x E X I we put Fj(x) = {f(x)). In this way we come to a canonical one-to-one correspondence between ordinary mappings f acting from X into Y and set-valued mappings
satisfying the condition (Vx E X)(card(F(x)) = 1). Evidently, there are various set-valued mappings canonically associated with the given ordinary mapping f : X -+ Y. For instance, we can define a set-valued mapping F! : Y -+ P ( X ) by the following formula:
Exercise 3. Show that there is a canonical one-to-one correspondence between ordinary mappings f acting from X into Y and set-valued mappings F : Y-+P(X) satisfying the condition that the family {F(y) : y E Y) forms a disjoint covering of X . Let {Xi : i E I) be a family of sets. In the usual way we define the union U{Xi : i E I) and the intersection n { x i : i E I) of this family. If J is an arbitrary subset of the set I, then {Xi : i E J) is called a subfamily of the family {Xi : i E I) (in fact, a subfamily of a given family is some restriction of the function which defines this family). We say that a family of sets {Xi : i E I) is disjoint if the equality
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holds for all indices i E I, j E I, i # j. We say that a family of elements {xi : i E I) is a selector of a family of sets {Xi : i E I) if the condition (Vi E I)(xi E Xi) holds. A selector {xi : i E I) is called injective if the corresponding function i-+xi ( i E I , xiEXi) is injective. The set of all selectors of a given family {Xi : i E I) is called the Cartesian product of this family and denoted by the symbol
Let X and Y be any two sets. Then the symbol Y X denotes the set of all mappings acting from X into Y. Obviously, the set yXcan be regarded as a particular case of the Cartesian product of a family of sets. The Axiom of Choice states, in fact, that the relation (Vi E I)(Xi
# 0)
implies
nixi
:
i E I) # 0.
Kelley showed that, in the theory Z F , the following two statements are equivalent: a) the Axiom of Choice; b) the product space of an arbitrary family of quasicompact topological spaces is quasicompact. Exercise 4, Prove in the theory Z F that statements .a) and b) are equivalent (we recall that a topological space E is quasicompact if any open covering of E contains a finite subcovering of E ) . Note that implication a ) S- b ) is widely known in general topology. In order to establish the converse implication, consider a family (Xi)ier of nonempty sets and take any element x such that
Further, for each index i E I, define
INTRODUCTION
and equip X,! with the topology
where
z= (0, xi,{x)). In this way, all Xi become quasicompact spaces. Finally, apply b) to the product space of the family ( X i ) ( E I , R e m a r k 1. A topological space is compact if it is quasicompact and Hausdorff simultaneously. At the present time, it is known that the following two assertions are not equivalent in ZF: a) the Axiom of Choice; b') the product space of an arbitrary family of compact spaces is compact. The countable form of the Axiom of Choice is the restriction of this axiom only t o countable families of sets:
If { X , : n E w ) as a n arbitrary countable family of n o n e m p t y pairwase disjoant sets, then there exists a selector of { X , : n E w ) . Obviously, the countable form of the Axiom of Choice is a very weak version of this axiom. It is not difficult to show, in the theory Z F , that: ( 1 ) D C implies the countable form of AC; (2) the countable form of AC is sufficient to prove the equivalence of the Cauchy and Heine definitions of a continuous function acting from R into R; (3) the countable form of AC implies that the union of a countable family of countable sets is a countable set; ( 4 ) the countable form of AC implies that any infinite set contains an infinite countable subset. Exercise 5. Prove all assertions (1) - ( 4 ) in the theory ZF Actually, the countable form of the Axiom of Choice is completely sufficient for classical mathematical analysis, classical Euclidean geometry and even for the elementary topology of point sets. Now, we are going to present some simple notions and facts from general topology (these facts will be needed in our further considerations). We recall that a topological space is any pair ( E , T) where E is a basic set and T is a topology (or a topological structure) defined on E. If T is fixed in our considerations, then we simply say that E is a topological space. Let E be a topological space and let X be a subset of E. We put:
CHAPTER
0
cl(X) = the closure of the set X ; int(X) = the interior of the set X ; bd(X) = the boundary of the set X . Consequently, closed subsets of E are all those sets X E E for which we have cl(X) = X , and open subsets of E are all those sets X C E for which we have int(X) = X . We say that a set X C E is an F,-subset of E if X can be represented as the union of a countable family of closed subse,ts of E. We say that a set X E E is a Ga-subset of E if X can be represented as the intersection of a countable family of open subsets of E. The Borel a-algebra of a space E is the a-algebra of subsets of El generated by the family of all open subsets of E. This a-algebra is denoted by the symbol B ( E ) . Obviously, we may say that B ( E ) is generated by the family of all closed subsets of E. Elements of B ( E ) are called Borel subsets of the space E. We say that a topological space E is a Polish space if E is homeomorphic to a complete separable metric space. We have the following. topological characterization of all Polish spaces: a topological space E is Polish if and only if it is homeomorphic to a G a subset of the Hilbert cube [0, l]'"(see, e.g., [89]). We say that a metriaable topological space E is an analytic (or Suslin) space if E can be represented as a continuous image of a Polish space. There is also another definition of analytic spaces starting with the so-called (A)operation applied to the family of all closed subsets of a Polish space (for the concept of (A)-operation and its basic properties, see e.g. [89] or [91]). Let X be a Polish topological space. The family of all analytic subsets of X is denoted by the symbol A(X). This family is closed under countable unions and countable intersections. Moreover, we have the inclusion
If a given space X is uncountable, then the inclusion mentioned above is proper (this classical result is due to Suslin; for the proof, see [89] or Chapter 1 of the present book). Another important result, due to Alexandrov and Hausdorff, states that any uncountable analytic set A C X contains a subset which is homeomorphic to the Cantor discontinuum. Hence the equality card(A) = c holds. In particular, for each uncountable Borel set B C X , we also have the equality card(B) = c (the proofs of these statements can be found in [89]). Let E be a topological space and let p be a measure given on E. We recall that p is a Bore1 measure (on E ) if
INTRODUCTION
17
We also say that p is a Radon measure on E if p is a-finite, dom(p) = B ( E ) and, for each Borel subset Z of E, the relation
holds true. Finally, we say that a Hausdorff topological space E is a Radon space if every a-finite Borel measure on E is Radon. According to the well-known result from topological measure theory, all Polish spaces turn out to be Radon spaces. More generally, any analytic space is Radon. In addition, if E is a Polish space and A is an arbitrary analytic subset of E, then the space E\ A is Radon, too. In other words, all coanalytic subsets of a Polish space are Radon. For more information about Radon measures and Radon spaces, see, e.g., [14]. These measures and spaces play an important role in various questions of analysis and probability theory. Let X and Y be any two topological spaces and let f be a mapping acting from X into Y. We say that f is a Borel mapping if, for each Borel subset Z of Y , the preimage f-l(Z) is a Borel subset of X. Clearly, every continuous mapping acting from X into Y is a Borel mapping. Also, the composition of Borel mappings is a Borel mapping. We say that a mapping f : X -+ Y is a Borel isomorphism from X onto Y if f is a bijection and both mappings f and f-' are Borel. In this case, we say that the spaces X and Y are Borel isomorphic, Obviously, if the spaces X and Y are homeomorphic, then they are Borel isomorphic. The converse assertion is not true in general. Let Z be a Borel subset of a Polish topological space, let Y be a metrizable topological space and let f be an injective Borel mapping acting from Z into Y. Then the image f ( 2 )is a Borel subset of Y , In particular, we see that the family of all Borel subsets of a Polish space E is invariant under the family of all injective Borel mappings acting from E into E. We have the following important result (see [89]). Let X and Y be any two uncountable Borel subsets of a Polish topological space E. Then there exists a Borel isomorphism from X onto Y. Since all infinite countable subsets of a Polish space E are Borel sets, being Borel isomorphic, we can easily deduce from the result mentioned above that, for any Borel subsets X and Y of the space E, these two conditions are equivalent: 1) card(X) = card(Y); 2) X and Y are Borel isomorphic. Unfortunately, we do not have (in the theory ZFC) an analogous nice equivalence for analytic subsets of Polish topological spaces.
CHAPTER 0
18
Notice also that if Z is a Borel subset of a Polish topological space and f is a Borel mapping acting from Z into a Polish space E, then the image f (2)is an analytic subset of E. The classical theory of Borel subsets and analytic subsets of Polish topological spaces is considered in detail in the monograph by Kuratowski [89] (see also [97], [91], [55], [56] and [65]). Let E be an arbitrary Polish topological space. We define the classes Pro(E), PTI( E ) ,
...,
Pr,(E),
...
of subsets of E by recursion. Namely, first of all, we put
Suppose now that, for a natural number n > 0, the class Pr,-I(E) has already been defined. If n is an odd number, then, by definition, Pr,(E) is the class of all continuous images (in E ) of sets from the class P T , - ~ ( E ) . If n is an even number, then, by definition, Pr,(E) is the class of all complements of the sets from the class P T , - ~ ( E ) . Finally, we put
Pr(E)= u{Pr,(E)
: n
< w).
Sets from the class P r ( E ) are called projective subsets of the space E. The notion of a projective set was introduced by Luzin and, independently, by Sierpiliski. At the present time, there are many remarkable works devoted to the theory of projective sets. Elements of this theory are presented in the monograph by Kuratowski mentioned above (a more detailed discussion of this subject can be found in [97], [91], [55], [56] and [65]). Thus we conclude that Borel subsets of E (i.e, sets from the class Pro(E)) and analytic subsets of E (i.e. sets from the class P r l ( E ) ) are very particular cases of projective sets. Note that many natural problems concerning projective sets cannot be solved in the theory Z F C . For example, the following statements are true: (a) it cannot be proved, in Z F C , that each uncountable set from the class P r a ( R ) contains a subset homeomorphic to the Cantor discontinuum; (b) it cannot be proved, in Z F C , that each set from the class P r 3 ( R ) is measurable in the Lebesgue sense. Note that the statement analogous to (b) and concerning the Baire property of sets from the class P r 3 ( R ) is true, too. Now, let us recall some elementary facts about the Baire property of subsets of general topological spaces. Let E be an arbitrary topological space.
I N T R O D U C TI O N
19
We say that a set X E E is nowhere dense (in E) if int(cl(X)) = 0. For example, if V is an open subset of E, then the set bd(V) is nowhere dense in E. We say that a set X E E is a first category subset of E if X can be represented in the form
where all X, (n E w ) are nowhere dense subsets of E. The family of all first category subsets of E is denoted by the symbol I((E). If E is not a first category space, then K ( E ) is a a-ideal of subsets of E. We say that a set X C E has the Baire property (in E ) if X can be represented in the form X=(UUY)\Z where U is an open subset of E and both Y and Z are first category subsets of E. It is easy to check that a set X E E has the Baire property if and only if X can be represented in the form X = V A P where V is an open subset of E and P is a first category subset of E. The family of all subsets of a space E, having the Baire property (in E ) , is denoted by the symbol Ba(E). Obviously, B a ( E ) is the a-algebra of subsets of E generated by the family T ( E ) U Ii'(E), where T ( E ) is the topology of E (i.e, the family of all open subsets of E). Hence we have the inclusion B(E) E Ba(E). As a rule, this inclusion is proper. But there are some interesting examples of topological spaces E for which this inclusion becomes the equality. For instance, if E is a classical Luzin subset of the real line R, everywhere dense in R, then we have K(E) = [ E ] S ~ , Consequently, in this case, we obtain B ( E ) = Ba(E). Extensive information on Luzin subsets of the real line is contained in [88], [89], [I101 and [117]. We shall deal with Luzin sets in the subsequent sections of our book. At this moment, we wish only t o notice that the existence of Luzin subsets of R cannot be proved in the theory ZFC (on the other hand, the existence of such subsets of R follows easily from the Continuum Hypothesis). Another interesting example (in the theory ZFC)of a topological space E for which the equality B(E) = Ba(E) holds can be obtained if we take the set of all real numbers equipped with the so-called density topology (see, e.g., [117]). We shall consider below some elementary properties of the density topology.
20
CHAPTER
0
Let E be again an arbitrary topological space and let X be a subset of E. We say that X has the Baire property in the restricted sense if, for each subspace Y of El the set X n Y has the Baire property in the space Y. Clearly, the family of all subsets of the space E, having the Baire property in the restricted sense, is a a-algebra of subsets of E. We denote this a-algebra by B a r ( E ) . Obviously, we have the inclusion
It is also easy t o check that B ( E ) E B a r ( E ) . Moreover, it can be shown that A(E) C Bar(E), i.e, all analytic subsets of E have the Baire property in the restricted sense (see [89]). Let X and Y be any two topological spaces and let f be a mapping acting from X into Y. We say that f has the Baire property if, for each Borel subset B of Y, the set f - l ( B ) has the Baire property in X . Evidently, every Borel mapping acting from X into Y has the Baire property. The composition of two mappings, each of which has the Baire property, can be a mapping without the Baire property.
Exercise 6. Give an example of two functions
each of which has the Baire property, but their composition g o f does not possess this property. However, if X , Y, Z are three topological spaces, f : X -+ Y has the Baire property, g : Y -+ Z is a Borel mapping and h : X -+ Z is the composition o f f and g, then h has the Baire property, too. In a similar way we can define a mapping with the Baire property in the restricted sense. Namely, we say that f : X -+ Y has the Baire property in the restricted sense if, for each Borel subset B of Y, the set f - l ( B ) is a subset of X having the Baire property in the restricted sense. All Borel mappings have the Baire property in the restricted sense. Let X and Y be any two topological spaces and let
be a set-valued mapping. We say that F has closed graph if
21
INTRODUCTION
is a closed subset of the product space X x Y. It is clear that if the given set-valued mapping F has closed graph, then, for each element x E X I the set F ( x ) is a closed subset of the' space Y. The converse assertion is not true in general. Set-valued mappings with closed graphs are important in different domains of mathematics, especially, in those questions which concern the existence of fixed points of set-valued mappings (we recall that a point x E dom(F) is a fixed point for a set-valued mapping F if x E F ( x ) ) . Note that theorems on the existence of fixed points for set-valued mappings found many interesting applications (see, e.g., [38] and [153]). Let X and Y be again two topological spaces and let
be a set-valued mapping. We say that F is lower semicontinuous if (1) for each point x E X, the set F ( x ) is closed in Y; (2) for each open subset V of Y, the set
is open in X . There are certain relationships between set-valued mappings with closed graphs and lower semicontinuous set-valued mappings (cf., for example, Theorem 1 below).
Exercise 7. Let X be a topological space and let
be a real-valued function on X. We recall that f is lower (respectively, upper) semicontinuous if, for any t E R, the set {x E X : f(x) > t ) (respectively, the set {x E X : f ( x ) < t)) is open in X . Show that: a) f is lower semicontinuous if and only if - f is upper semicontinuous; b) f is lower semicontinuous if and only if the set
is closed in the product space X x R; c) f is lower semicontinuous if and only if, for each xo E X , we have the inequality liminf f (x) 1 f (go);
.+,,
d) f is continuous if and only if it is lower and upper semicontinuous.
CHAPTER
Exercise 8. Let
0
X be a set and let
be a real-valued function on
X .We introduce two set-valued mappings
by the following formulas:
Suppose now that X is a topological space. Show that the following two assertions are equivalent: a) f is lower semicontinuous (as an ordinary function); b) F 1 , j is lower semicontinuous (as a set-valued function). Show that the next two assertions are also equivalent: c) f is upper semicontinuous (as an ordinary function); d) F 2 , j is lower semicontinuous (as a set-valued function). Exercise 9. Let
X be a nonempty quasicompact topological space and
let
f : X+R be a ldwer semicontinuous function. Show that there exists a point go E X satisfying the relation f (xo) = i n S c ~ x(.I.f Formulate and prove an analogous result for upper semicontinuous realvalued functions defined on X. Exercise 10. Let
X and Y be any two topological spaces and let
be any two lower (respectively, upper) semicontinuous functions such that
Define a function h : XxY-,R
by the formula
23
INT R O D U C TIO N
Show that h is also lower (respectively, upper) semicontinuoue. Exercise 11. Let X be a completely regular topological space and let
be a lower semicontinuous function such that f (x) 2 0 for all x E X . Show ~ functions from X into R,satisfying the that there exists a family ( f i ) $ €of following conditions: 1) for each i E I, the function fi ie continuous; 2) for all i E I and for all x t! X , we have fi(x) 0; 3) f = supi€rfi; 4) card(I) 5 w(X) w where w(X) denotes the topological weight of X (i.e. w(X) is the smallest cardinality of a base of X ) . In particular, if X has a countable base, then f can be represented as a pointwise limit of an increasing sequence of positive continuoue real-valued functions defined on X .
>
+
Exercise 12. Let [a, b] be a closed subinterval of R and let
>
be two functions such that f g. Suppose, in addition, that f is lower semicontinuous and g is upper semicontinuoue. Demonstrate that there exists a continuoue function
satisfying the inequalities g 5 h S f This simple result admits a number of generalizations and, actually, is a direct consequence of the well-known Michael theorem on continuous selectors (see [105], [I061 or [122]). Exercise 13. Let X be a second category topological space and let {fi : i E I) be a family of real-valued lower semicontinuoue functions on X . Suppose, in addition, that, for each point x E X , the eet {fi(x) : i E I) is bounded from above. Show that there exists a nonempty open set V E X for which the set u{fi(V) : i E I) is bounded from above, too. Formulate and prove an analogous result for upper semicontinuous functions.
24
CHAPTER
0
Exercise 14. Let (GI+) be a second category topological group and let {fi : i E I} be a family of real-valued lower semicontinuous functions on G. Suppose that the following conditions hold: a) for each index i E I, the function fa is subadditive, i.e, we have
b) for each point x E G, the set {fi(x) : i E I ) is bounded from above. Show that the given family { f i : i E I ) is locally bounded from above. This means that, for any point x E G, there exists a neighbourhood V(x) for which the set u{fi(V(x)) : i E I ) is bounded from above. Formulate and prove an analogous statement for upper semicontinuous functions. Notice that the result presented in Exercise 14 easily implies the wellknown Banach-Steinhaus theorem (see [8] or [63]). For our further considerations, we need one auxiliary proposition on closed projections. This proposition is due to Kuratowski (see, for instance, [89]) and has numerous applications. Lemma 1. Let X be a topological space, let Y be a quasicompact space and let prl denote the canonical projection from X x Y into X , i.e. the mapping prl : X x Y - + X
is defined b y the formula
Then prl is a closed mapping, i.e. for each closed subset A of X x Y , the image prl(A) is closed in X . Proof. Take any point x E X such that U n prl(A) # 0 for all neighbourhoods U of x. We are going to show that x € prl(A). For this purpose, it is sufficient to establish that
Suppose otherwise, i.e. ({x} x Y ) f l A = 0. Then, for each point y E Y , there exists an open neighbourhood W((x, y)) of the point (x, y), satisfying the relation w ( ( x l Y)) n = 0.
e
INTRODUCTION
We may assume, without loss of generality, that
where U(x) is an open neighbourhood of x and V(y) is an open neighbourhood of y. Since the space {x} x Y is quasicompact, there exists a finite sequence ( x , Y ~ )(x,Yz), , .,* , (x, ~ t a ) of points from {x} x Y, such that {W((x, ys)) : 1 i n} is a finite covering of {x) x Y. Now, let us put
< <
0 ( x ) = n{Ui(x) : 1
< i j n).
Then it is easy to check that O(x) is a neighbourhood of the point x, satisfying the equality O(x) n prl(A) = 0. But this is impossible. So we get a contradiction, and the Kuratowski lemma is proved. Exercise 15. Show, e.g, for a metrizable topological space Y, that the property of Y described in Lemma 1 is equivalent to the compactness of Y. More precisely, prove that, for a given metric space Y, the following two assertions are equivalent: a) Y is compact; b) the canonical projection
is a closed mapping. Now, we want to give some applications of Lemma 1 to set-valued mappings. For the sake of simplicity (and motivated by the aims of mathematical analysis), we restrict our further considerations to the class of all metric spaces, but it is not difficult to see that the results presented below remain true in more general situations. Theorem 1. Let X be a metric space, let Y be a compact metric space P(Y) be a set-valued mapping. Then the following two and let F : X assertions are equivalent: 1) F has closed graph; 2) for each point x E X , the set F ( x ) is closed an Y and, for each closed subset A of Y, the set -(
26
CHAPTER 0
is closed in X Proof. Suppose that assertion 1) is true. Then, obviously, for any element x E X , the set F ( x ) is closed in Y. Now, let A be an arbitrary closed subset of Y . It is easy t o see that
Clearly, the set ( X x A ) n G F is closed in the product space X x Y , as the intersection of two closed subsets of this space. Therefore, by the Kuratowski lemma proved above, we obtain that F-l(A) is a closed subset of the space X . Thus assertion 2) is true and implication 1) + 2) is established. Now, assume that assertion 2) holds. Let us prove that GF is a closed subset of the product space X x Y. For this purpose, take an arbitrary sequence {(xi, yi) : i E N} of points of the graph GF, such that
Let us show that the point (2, y) also belongs t o GF. Suppose otherwise, i.e. (x, y) 6 GF, This means, by the definition, that y 6 F ( x ) . Since F ( x ) is a closed set in Y , there exists a neighbourhood V(y) C Y of the point y, satisfying the relation V(y) n F ( x ) = 0. Furthermore, we have
Without loss of generality, we may assume that yi E V(y) for all indices i E N. Let us put A = { y ) u { y i : EN). Evidently, A is a closed subset of Y. It is also clear that A n F ( x ) = 0. On the other hand, we have
Taking into account that bimi,+, closed in X , we get
xi = x , and that the set F-l(A) is
which yields a contradiction. It shows us that our set-valued mapping F has the closed graph. Hence implication 2) + 1) is established, and the proof of Theorem 1 is completed.
Remark 2. One can easily see that, in the proof of Theorem 1, only implication 1) + 2) relies essentially on the Kuratowski lemma on closed
INTRODUCTION
27
projections. The converse implication 2) j 1) does not need this lemma. So we can conclude the following fact. Suppose that X and Y are arbitrary metric spaces and F : X 4 P ( Y ) is a set-valued mapping satisfying the conditions: a) F(x) is a closed subset of Y for each point x E X ; b) F - l ( A ) is a closed subset of X for each closed set A S Y . Then the set-valued mapping F has closed graph. A particular case of this fact is the following one. Let X and Y be any two metric spaces and let f : X --r Y be a continuous mapping. Then the graph
of this mapping is a closed subset of the product space X x Y. Notice that the converse assertion is not true in general. Indeed, it is not difficult to construct an example of a function g : R -+ R which is discontinuous but has closed graph. Moreover, there exists a function g : R 4 R with closed graph, such that the set of all points of discontinuity of g is a nonempty perfect subset of R (the function g with this property can be constructed by starting with the classical Cantor set on the real line). On the other hand, it is reasonable to notice here that any function f : R -r R having closed graph belongs to the first Baire class. Therefore, according to the well-known Baire theorem (see, e.g., [ I l l ] or [117]), for each nonempty perfect set A C R , there exists a point of A at which the function f lA is continuous. Exercise 16. Let X be a metric space, let Y be a compact metric space and let f be a mapping acting from X into Y. Deduce from Theorem 1 that the following two conditions are equivalent: a) f is a continuous mapping; b) the graph G j is a closed subset of the product space X x Y . Exercise 17. Let X and Y be compact metric spaces and let f be a mapping acting from X into Y . Deduce from the result of the previous exercise that these two conditions are equivalent: a) f is a continuous mapping; b) the graph Gf is a compact subset of the product space X x Y . Let us now present a typical application of the Kuratowski lemma to the classical theory of real functions. We mean here the existence of continuous nowhere differentiable functions. Let C[O, 11 denote, as usual, the Banach space of all continuous real-valued functions defined on the unit segment
28
C H A PT ER
0
[O, 11. The following famous result is due to Banach and Mazurkiewicz (see [6] and [104]). T h e o r e m 2. The family of all those functions from the Banach space CEO, 11) which a n nowhere differentiable on [O, 11, is the complement of a first category set. Proof. Let h be a nonzero rational number such that Ihl natural number n > 0, consider the set
< 1. For every
I(f (x + 6) - f (x))lsl In)I. It is not hard to check that a h , , is a closed subset of the space C[O, 11. Indeed, let us put
l(f (x -t 6) - f (2))/61 5 741. Then
Zh,n
is a closed subset of the product space C[O,11 x [0, 11 and
where pr1 : C[O, 11 x [0, 11 -+ C[O, 11
denotes the canonical projection onto C[O,11, Taking account of the compactness of the unit'segment [O, 11 and applying Lemma 1, we immediately obtain that the set Q h l n is closed in C[O,11. Simultaneously, Q h l n is nowhere dense in C[O, 11 (the latter fact is almost trivial from the geometrical point of view). Consequently, the set
is of first category in C[O, 11. Now, it is clear that any function belonging to the set C[O,11 \ D is nowhere differentiable on [ O , l ] . This completes the proof of Theorem 2. Exercise 18. Let f be a function acting from R into R and let x be a point of R.Recall that f possesses a symmetric derivative at the point x if there exists a (finite) limit
INTRODUCTION
29
In such a case, this limit is called the symmetric derivative off at x (denoted by the symbol f:(x)). Demonstrate that f can possess a symmetric derivative a t a point x being even discontinuous at this point. Show that if f is differentiable (in the usual sense) at a point x, then there exists a symmetric derivative fL(x) and the equality
is fulfilled. Show also that the converse assertion is not true in general. Does an analogue of Theorem 2 hold for the symmetric derivative (instead of the derivative in the usual sense)? The notion of a symmetric derivative of a function can be regarded as a simple example of the concept of a generalized derivative. In the subsequent sections of the book we shall discuss some other types of a generalized derivative. The notion of an approximate derivative (introduced by Khinchin in 1914) is of special interest and will be defined and discussed in Chapter 4. It is well known that this notion plays an important role in various questions of real analysis (for instance, in the theory of generalized integrals). The definition of an approximate derivative relies on the concept of a density point for a given Lebesgue measurable subset of R. Let X denote the standard Lebesgue measure on R and let X be an arbitrary X-measurable subset of R. We say that x E R is a density point for X if The classical theorem of real analysis, due to Lebesgue, states that almost all (with respect to A) points of X are its density points. In order to establish this fact, we need the concept of Vitali covering of a set lying in R , and the important result of Vitali concerning such coverings. For the sake of completeness, we formulate and prove this result. Let {Di : i E I) be a family of nondegenerate segments on R and let Z be a subset of R. We say that this family is a Vitali covering of Z if, for each point r E Z , we have
in f {X(Di) : i E I, r E Dd) = 0. The following fundamental result was obtained by Vitali (cf., e.g., [ l l l ] , [I171 or [127]).
Theorem 3. If Z is a subset of R and {Da : i E I ) is a Vitali covering of Z , then there exists a couniable set J C I such that ihe padial family {Dj : j E J) is disjoint and X ( z \ u { D j : j E J ) ) = 0.
CHAPTER 0
30
Proof. Without loss of generality we may assume that Z is bounded. Let U be an open bounded set in R containing Z . We may also assume that Di C U for each index i E I. Define by recursion a disjoint countable subfamily of segments
Take Dqo) arbitrarily. Suppose that Dqo), Di(l), been defined. Put
t ( k ) = sup{X(Di) : Di C U \ (Da(o)U
... , u u -
Dqlc) have already
U Di(k))).
Let Di(k+l) be a segment from {Di : i E I) such that
In this way we obtain the desired disjoint sequence {Di(k) : k E N ) . Note that
so we have, in particular,
We are going to show that
is the required subfamily. For this purpose, denote by Dl(k) the segment in R whose centre coincides with the centre of Dqk) and for which
Let us demonstrate that, for each natural number n, the inclusion Z \ u { D ~ ( ~: ) k E N ) 5 u{D:(~) : k E N , k
> n}
holds true. Indeed, let z be an arbitrary point from Z\U{Dd(k) : k E N}. Then, in particular,
Since {Da : i E I } is a Vitali covering of Z , there exists a segment Di for which E Di, Di fl (Di(0) U --.U Dd(n))= 0,
31
INTRODUCTION
Obviously, we have A(Di) > 0. At the same time, as mentioned above, limk-++ooA(Di(k)) = 0. So, for some natural numbers k, we must have
Let k be the smallest natural number with this property. Evidently, k Thus we get
> n.
In addition, V D i ) S 2X(Di(k+1))1 which immediately implies (in view of the definition of Dj(k+l)) the inclusion Di E D5(k+,). Consequently,
Finally, since, for each natural n , we have
we conclude that
A(Z \ U{Dj : j E J ) ) = 0,
and the theorem is proved. We shall present some standard applications of Theorem 3 in the subsequent sections of the book. Here we only recall how the above-mentioned Lebesgue result on density points of A-measurable sets can be easily derived from Theorem 3.
Theorem 4. Let X be an arbitrary A-measurable set on R and lei d(X) = {x E R : x is a density point of X}.
Then we have
A(X \ (X
n d ( x ) ) ) = 0.
Proof. We may assume, without loss of generality, that X is bounded. For any natural number n > 0, let us define
32
CHAPTER
0
Clearly, it suffices to show that A*(Xn) = 0 where A* denotes the outer measure associated with A. For this purpose, fix E > 0. Let U be an open subset of R such that
In virtue of the definition of X, , there exists a Vitali covering {Di : i E I } of X, such that (Vi E I)(A(Di n X)/X(Di)
< 1 - lln).
Obviously, we may suppose that Di 5 U for each index i E I. According to Theorem 3, there exists a disjoint countable subfamily {Dj : j E J) of this covering, for which we have A(X,
\ U{Dj
: jE
J}) = 0.
Then we can write
5 (1 - l/n)A(U) Since
E
< (1 - l/n)(X*(X,) + E ) .
> 0 was taken arbitrarily, we have
Finally, in view of the inequality A*(X,) 0, and the theorem is proved.
< +co,we conclude that A*(Xn) =
Exercise 19. For any A-measurable set X C R, show that the set d(X) is Bore1 in R. Exercise 20. Demonstrate that if z E R is a density point of two A-measurable sets X and Y , then z is a density point of the set Z = X n Y . This fact is important for introducing the so-called density topology on R which will be discussed in our further considerations.
1. Cantor and Peano type functions
It is well known that one of the first mathematical results of Cantor (which turned out to be rather surprising to him) was the discovery of the existence of a bijection between the set R of all real numbers and the corresponding product set R 2 = R x R (i.e, the Euclidean plane). For a time, Cantor did not believe that such a bijection exists and even wrote to Dedekind about his doubts in this connection. Of course, Cantor already knew of the existence of a bijection between the set N of all natural numbers and the product set N x N. A simple way to construct such a bijection is the following one. First, we observe that a function
defined by the formula
is a bijection between N and the set of all strictly positive natural numbers. Then, for each natural n > 0, we have a unique representation of n in the form n = 2k (21 + 1) where k and I are some natural numbers. Now, define a function
by the formula g(n) = (k,1 ) (n E N \ { O H . One can immediately check that g is a bijection, which also yields the corresponding bijection between N and N x N. By starting with the latter bijection, it is not hard to establish a one-toone correspondence between the real line and the Euclidean plane (respectively, between the unit segment [0, 11 and the unit square [O,112). Indeed, a simple argument (in ZF) shows that the sets
'
34
CHAPTER 1
are equivalent, i.e, there exists a bijective mapping from each of them to any other one. So we only have to check that the sets
are equivalent, too. But this is obvious since the product set 2N x 2N is equivalent with the set 2NxN and the latter set is equivalent with 2N because of the existence of a bijection between N and N x N . Keeping in mind these simple constructions, it is reasonable to introduce the following definition. We say that a mapping f acting from R into R 2 (respectively, from [O, 11 into [O, 112) is a Cantor type function if f is a bijection. As mentioned above, Cantor type functions do exist.
Remark 1. As pointed out earlier, one-to-one correspondences between N and N x N (respectively, between R and R x R or between [0, 11 and [O, 112) can be constructed effectively, i.e, without the aid of the Axiom of Choice. In this connection, let us recall that, for an arbitrary infinite set X , we also have a bijection between X and X x X , but the existence of such a bijection needs the whole power of the Axiom of Choice. More precisely, according to the classical result of Tarski (cf. [91]), the following two assertions are equivalent in the theory ZF: 1) the Axiom of Choice; 2) for any infinite set X , there exists a bijection from X onto X x X. Exercise 1. Let X be an arbitrary set. Show, in the theory Z F , that there exists a well ordered set Y such that there is no injection from Y into X . We may suppose, without loss of generality, that X n Y = 0. Demonstrate (in the same theory) that if card(X x Y) 5 card(X U Y), then there exists an injection from X into Y and, consequently, X can be well ordered. Show also (in ZF) that the relation
implies the inequality card(X x Y)
< card(X U Y).
CANTOR A N D PEANO T Y P E FUNCTIONS
35
Deduce from these results that, in ZF, the following two assertions are equivalent: 1) the Axiom of Choice; 2) for any infinite set X , the equality
is satisfied. Now, let f be an arbitrary Cantor type function acting, for example, from R onto R2. It is well known that, in such a case, f cannot be continuous. Indeed, suppose for a moment that f is continuous. Then we may write R2= U{f ([-n, n]) : n E N ) where each set f([-n,n]) (n E N ) is compact (hence closed) in R2.In accordance with the classical Baire theorem, at least one of these sets has a nonempty interior. Let k be a natural number such that
Then we have a bijective continuous mapping
which obviously is a homeomorphism between [-k, k] and f ([-k, k]). But this is impossible since [-k, k] is a one-dimensional. space and f([-k, k]) is a two-dimensional one. If we want to avoid an argument based on the notion of a dimension of a top.ologica1space (and it is reasonable to avoid here such an argument because we do not; discuss this important notion in our book), we can argue in the following manner. Consider the function
which also is a homeomorphism. Let L denote any circle contained in the set f ([-k, k]), i.e. let L be a subset of f ([-k, k]) isometric to
where r is some strictly positive real (the existence of L is evident since f([-k, k]) has a nonempty interior). We thus see that the function
36
CHAPTER 1
is injective and continuous. This immediately yields a contradiction since there is no injective continuous function acting from a circle into the real line (cf, the next exercise).
Exercise 2. Let L be a circle on the plane and let g : L --,R be a continuous mapping. By using the classical Cauchy theorem on intermediate values for continuous functions, prove that there exist two points z E L and z' E L satisfying the relations: a) g(z) = g(z'); b) z and z' are antipodal in L, i.e, the linear segment [z, z'] is a diameter of L. In particular, g cannot be an injection. This simple result admits an important generalization to the case of an n-dimensional sphere (instead of L) and of an n-dimensional Euclidean space (instead of R).The corresponding statement is known as the BorsukUlam theorem on antipodes and plays an essential role in algebraic topology (see, for example, [90]). In particular, this theorem shows that there are no injective continuous mappings from the sphere Sn into the space R n. The following statement is also of some interest in connection with Cantor type functions (see, e.g., [134]).
Theorem 1. Let f be a function from R 2 anto R continuous with respect to each of the variables x and y (se parately). Then f as not an injection. Proof. Suppose otherwise, i.e, that our f is injective. Denote
Then, according to the assumption of the theorem, tion from R into R.Let us put
+ is a continuous func-
Since f is injective, we have a # b. Consequently, either a < b or b < a. We may assume, without loss of generality, that a < b. The function $, being continuous on the segment [ O , l ] , takes all values from the segment
In particular, there exists at least one point xo E ]O,l[ such that
CANTOR AND PEANO T Y P E FUNCTIONS
Further, let us define
Then $ is a continuous function, too, and
Hence we have the inequalities
which imply the existence of a neighbourhood U(0) of the point 0, such that (VY E U(O))(a < +(Y)< b) or, equivalently, (VY E U(O))(a
< f(.o,Y) < b).
Of course, we may additionally suppose that U(0) is contained in ] - 1, I[. Thus, on the one hand, we have the inclusion
On the other hand,
so, for some reals yo
# 0 and
XI,
we get
which contradicts the injectivity of f . The contradiction obtained finishes the proof of Theorem 1. E x e r c i s e 3. Does there exist an injective mapping
such that f is continuous with respect t o one of the variables x and y? E x e r c i s e 4, Show that there exists a bijection
f
: [O, 11 -, [o, 112
CHAPTER
such that the function
prl o
1
f is continuous, where
denotes, as usual, the first canonical projection from [O, 112 onto [ O , l ] . We thus see that Cantor type functions cannot be continuous. In this connection, it is reasonable to ask whether there exist continuous surjections from R onto R 2 or from [0, 11 onto [O, 112. It turned out that such surjections do exist and the first example of the corresponding function acting from [O, 11 onto [O, 112 was constructed by Peano. Hence the following definition seems to be natural. Let
We shall say that f is a Peano type function iff is continuous and surjective. In order to demonstrate the existence of Peano type functions, we recall the classical Cantor construction of his famous discontinuum. Take the unit segment [O,1]on the real line R . The first step of Cantor's construction is to remove from this segment the open interval ]1/3,2/3[ whose centre coincides with the centre of [ O , l ] and whose length is equal to the one-third of the length of our segment. After this step we obtain the two segments without common points. Then we apply the same operation to each of these two segments, etc. After w-many steps we come to the subset C of [0, 11 which is called the Cantor discontinuum (or the Cantor space). The set C is closed (since we removed open intervals from [0, 11) and, in addition, C is perfect because the removed intervals are disjoint and pairwise have no common end-points. Moreover, since the sum of lengths of the removed intervals is equal to 1 (which can easily be checked), we infer that C is nowhere dense in R and its Lebesgue measure equals zero. Consequently, C is a small subset of R from the point of view of the Baire category and from the point of view of the standard Lebesgue measure A on R. The geometric construction of C described above and due to Cantor himself is rather visual but, sometimes, other constructions and characterizations of C are needed in order to formulate the corresponding results in a more general form. We present some of such constructions and characterizations of Cantor's discontinuum in the next two exercises.
Exercise 5. Take the two-element set 2 = {0,1) and equip this set with the discrete topology. Equip also the Cartesian product 2" with the product topology. Demonstrate that 2W is homeomorphic to the classical Cantor discontinuum C.
CANTOR A N D P E A N O T Y P E FUNCTIONS
39
Exercise 6. Let E be a topological space. Show that E is homeomorphic to C if and only if the conjunction of the following four relations holds: a) E is nonempty and compact; b) E has a countable base; c) there are no isolated points in E; d) E is zero-dimensional, i.e. for each point e E E and for any neighbourhood U(e) of el there exists a neighbourhood V(e) of e such that
where the symbol bd(V(e)) denotes the boundary of V(e). Actually, the last relation means that the family of all clopen subsets of E forms a base for E. The abstract characterization of the Cantor space, given in Exercise 6, implies many useful consequences. For instance, by using this characterization, it is not difficult to show that, for each natural number k 2 2, the product space k W is homeomorphic to the Cantor discontinuum (of course, here k is equipped with the discrete topology). Exercise 7. Demonstrate that k W is homeomorphic to C. Verify also that w W is homeomorphic to the space of all irrational numbers (where w is equipped with the discrete topology). Naturally, the Cantor discontinuum ha.s numerous applications in various branches of mathematics (especially, in topology and analysis). The next exercise presents a typical application of C in real analysis. Exercise 8. Construct a set C1on [O,1] such that: 1) C' is homeomorphic to C; 2) the Lebesgue measure of C1is strictly positive. Deduce from relations 1) and 2) that the measure A is not quasiinvariant with respect to the group of all homeomorphisms of R, i.e. this group does not preserve the u-ideal of all A-measure zero sets. It immediately follows from the construction of C that, for each clopen set X g C and for any E > 0, there exists a finite partition of X consisting of clopen subsets of X , each of which has diameter strictly less than E . It is also easy to check that every zero-dimensional compact metric space possesses an analogous property. At the same time, if E is an arbitrary
40
CHAPTER 1
compact metric space, then, for each closed set X E E and for any E > 0, there exists a finite covering of X consisting of closed subsets of X , each of which has diameter strictly less than E . These simple observations lead to the following important statement due to Alexandrov.
Theorem 2. Lei E be an arbitrary nonempty compact metric space. Then there exasts a continuous surjection from the Cantor space C onto E. Proof. Taking account of the preceding remarks, we can recursively define two sequences
satisfying the conditions: 1) for any n E w, the finite family
is a partition of C consisting of clopen subsets of C each of which has diameter strictly less than l / ( n 1); 2) for any n E w, the finite family
+
is a covering of E by nonempty closed sets each of which has diameter strictly less than l / ( n 1); 3) for any n E w, the family (Xn+l,k)llklm(n+l) (respectively, the family (Yn+l,k)lSklm(n+l)) is inscribed in the family (Xn,k)llk
+
we have xn+1,1
s x n , k * Yn+l,~s y n , k *
Let now x be an arbitrary point of C . Then x uniquely determines the sequences {k(n) : n E w ) , {Xn,k(n) : n E w } ,
C A N T O R AND PEANO T Y P E FUNCTIONS
such that (Vn E ~ ) (Il k(n)
I m(n) & x E Xn,k(n)).
Consider the corresponding sequence {Yn,k(n) : n E w ) . Obviously, we have (Vn E w)(Yn+l,k(n+l) E Yn,k(n)), limn,+,
diam(ynpk(,))= 0.
Hence there exists one and only one point y belonging to all the sets Yn,lc(n)(n E w). Let us put y = f(x). In this way we obtain the mapping
f :C-+E. By starting with the definition of f , it is not hard to verify that f is continuous and surjective. Theorem 2 has thus been proved.
Exercise 9. Let X be a nonempty closed subset of C.Show that there exists a continuous mapping
satisfying the relation (Vx E X ) ( f ( x ) = x ) . In other words, each nonempty closed subset of C is a retract of C. Notice that this simple but useful result follows directly from the well-known Michael theorem concerning the existence of continuous selectors for lower semicontinuous set-valued mappings defined on zero-dimensional paracompact spaces (see [105], [I061 or [122]).
Exercise 10. Using the result of Exercise 9, give another proof of Alexandrov's theorem. Namely, starting with a canonical continuous surjection h : 2W-+ [O,l], show that there exists a continuous surjection hl : 2W+ [O, 1IW Then, for each closed subset Y of [0, 1IW,show that there exists a closed subset X of C such that hl(X) = Y. Finally, apply the classical theorem of Urysohn stating that every compact metric space can be realized as a closed subset of the space [0, 1IW.
42
C HA P T E R 1
Theorem 2 immediately implies the existence of Peano type functions. T h e o r e m 3. There exists a Peano type function, i.e. there exast a contanuous surjection from [O,l] onto [0, 11'. Proof. It follows from Theorem 2 that there is a continuous surjection
Consider any open interval U that was removed from [0, 11 during the construction of C. We may add U to the domain of g and extend g in such a way that the extended function will be linear on U and will coincide with g on the end-points of U. Doing this for all removed intervals simultaneously, we come to the function f defined on the whole segment [0, 11. The construction of f immediately implies that f is continuous. In addition, since f is an extension of g , we conclude that f is a surjection as well. Furthermore, let us note that the existence of a Peano type function from [O, 11 onto [O, 112 implies at once the existence of a continuous surjection
Such a function h can also be regarded as a Peano type function (acting from R onto R 2 ) . Exercise 11. Let P be a nonempty perfect subset of the real line. Demonstrate that there exists a continuous surjection from P onto the square [O, 112. Infer from this result that there exists a disjoint family
of nonempty perfect subsets of P, such that card(J) = c where c denotes, as usual, the cardinality of the continuum. Exercise 12. Prove that, for any Peano type function
the relation
{ x E R : f { ( x ) exists}U { x E R holds true (cf. Exercise 7 from Chapter 10).
:
f i ( x ) exists} # R
CANTOR AND PEANO T Y P E FUNCTIONS
43
Exercise 13. By starting with the existence of a Peano type function, demonstrate that there exists an injective continuous mapping
such that the orthogonal projection of g([O, 11) on the plane R x R x (0) coincides with the unit square [O, 112. Thus we get in the space R3 the simple arc g([O, 11) (which is a homeomorphic image of [O,l]) whose orthogonal projection on the above-mentioned plane has nonempty interior (with respect to that plane). As shown by Theorem 2, any nonempty compact metric space is a continuous image of C or, equivalently, of 2 W . A natural question arises concerning the description of all those Hausdorff topological spaces which are continuous images of C. It turns out that only nonempty compact metrizable spaces are such images. The following exercise provides the corresponding result. Exercise 14. Let E be a topological space. We say that a family S of subsets of E is a net in E (in the sense of Archangelski) if, for any open set U C E , there exists a subfamily of S whose union coincides with U. Clearly, every base of E is a net but the converse assertion does not hold in general. Let now E be a compact space and let n be an infinite cardinal number. Show that if E admits a net of cardinality n, then E possesses also a base of cardinality K . Deduce from this fact that if E' and E" are two compact spaces and g : El -r E" is a continuous surjection, then the topological weight of E" is less than or equal to the topological weight of El. In particular, if El has a countable base, then E" has a countable base, too. Conclude from this result that if a Hausdorff topological space Y is a continuous image of C , then Y is compact and metrizable. Exercise 15. Let K be an arbitrary infinite cardinal number. Equip the set 2' with the product topology (where the two-element set 2 = {0,1) is endowed with the discrete topology). The space 2' is usually called the generalized Cantor discontinuum (of weight n). This space can also be regarded as a commutative compact topological group with respect to the addition operation modulo 2. Hence there exists a Haar probability measure ,u on 2 # . By starting with the fact that ,u(U) > 0 for each nonempty
CHAPTER
44
1
open subset U of 2', show that 2' satisfies the so-called Suslin condition or countable chain condition, i.e, any disjoint family of nonempty open subsets of 2' is at most countable. Deduce from this fact that if a topological space E is a continuous image of 2') then E satisfies the countable chain condition, too. Conclude that there exists a nonempty compact topological space X such that, for all n 2 w , there is no continuous surjection from 2' onto X . On the other hand, show that, for any compact topological space Y of weight n, there exists a closed subset Z of 2' such that Y is a continuous image of Z . Note that the results presented in Exercise 15 are essentially due to Szpilrajn (Marczewski). As mentioned earlier, the Cantor set C is small from the topological point of view (i.e. C is a nowhere dense subset of the real line R) and from the measure-theoretical point of view (i.e. C is of Lebesgue measure zero). On the other hand, the operation of vector sum yields the set
which is not small at all. Indeed, it can easily be seen that this set contains in itself a nonempty open interval. More precisely, we have the equality
+
Exercise 16. Prove that C C = [0,2]. Moreover, give a simple geometrical interpretation of this equality. The result of Exercise 16 shows, in particular, that the operation of vector sum does not preserve the two classical u-ideals on the real line: the u-ideal K ( R ) of all first category subsets of R and the u-ideal I(A) of all Lebesgue measure zero subsets of R. Let us note that, at the same time, there exist many u-ideals on R which are (under some additional settheoretical axioms) isomorphic to Ii'(R) and I(A), invariant with respect to the group of all translations of R and also invariant with respect to the operation of vector sum of sets. Exercise 17. Consider on the Euclidean plane R 2 the family of all straight lines parallel to the line (0) x R. Let J denote the a-ideal of subsets of the plane, generated by this family. Check that: a) 3 is invariant under the group of all translations of R ~ ; b) J is invariant under the operation of vector sum of subsets of R 2 .
45
CANTOR A N D PEANO T Y P E FUNCTIONS
In addition, note that, by assuming the Continuum Hypothesis (or, more generally, Martin's Axiom), it can be shown that J is isomorphic to each of the a-ideals IC(R2) and I(X2) where X2 denotes the standard twodimensional Lebesgue measure on R 2 . This fact follows directly from the Sierpiliski-Erdos Duality Principle (see, for instance, [117], [I101 or [26]). Let us also remark that the a-ideals K ( R ) and (respectively, I(A) and I()r2)) are isomorphic to each other, and the corresponding isomorphisms can be constructed within the theory ZF & DC (see, e.g., [26]). Now, by taking account of the fact that the additive group of R is isomorphic to the additive group of R 2 , demonstrate that there exists a a-ideal I of subsets of R , satisfying the relations: (a) I is invariant with respect to the group of all translations of R ; (b) I is invariant with respect to the operation of vector sum of subsets of R ; (c) under the Continuum Hypothesis (or, more generally, under Martin's Axiom), I is isomorphic to each of the a-ideals K ( R ) and I()r). The preceding considerations show us that the operation of vector sum is rather bad from the point of view of preserving a-ideals on R . Analogously, this operation is bad from the point of view of descriptive set theory. The latter fact may be illustrated, for instance, by various kinds of examples of two Gpsubsets of R whose vector sum is not a Borel subset of R (on the other hand, such a vector sum is always an analytic subset of R ) . A similar phenomenon can be observed when dealing with the distance set and with the difference set of a given point set lying on the real line or in a finite-dimensional Euclidean space. We recall that the distance set of a set X lying in a metric space ( E , d) is the set of all distances d(x, y) where x and y range over X. The difference set of a set X lying in a commutative group +) is the set of all elements x y where x and y range over X. Sierpiliski was the first mathematician who gave an example of a Gasubset of the Euclidean plane, whose distance set is not Borel (see [139]). Much later, several authors (see e.g. [41], [123], [I471 and [148]) constructed two Gpsubsets of the real line, whose vector sum is not Borel. Moreover, Erdos and Stone [41] constructed a compact subset of R and a Ga-subset of R , such that their vector sum is not a Borel set in R . Here we only wish to present an example of a Ga-subset of R whose difference set is not Borel. This example is due to Rogers (see [123]).
-
(r,
First of all, we need to recall the notion of the Hausdorff metric. Let ( E , d) be an arbitrary metric space. Denote by the symbol F(E)the family
46
C H A PTER 1
of all nonempty closed bounded subsets of E . For any two sets X E F ( E ) and Y E F ( E ) , we define
where V, ( X ) (respectively, V, (Y)) denotes the E-neighbourhood of X (respectively, of Y). It can easily be seen that d' is a metric on F ( E ) , so we get the metric space ( F ( E ) , dl). The metric d' is usually called the Hausdorff metric associated with d. If the original metric space ( E , d) has good properties, then, sometimes, those properties can be transferred to the space ( F ( E ) , d'). For instance, 1) if ( E , d) is complete, then ( F ( E ) , dl) is complete, too; 2) if ( E , d) is compact, then (F(E),dl) is compact, too. Assertions 1) and 2) can be established without any difficulties. On the other hand, let us remark that if the original space ( E , d) is separable, then, in general, ( F ( E ) , d f ) need not be separable. Moreover, it can easily be observed that if ( E , d) is bounded but not totally bounded, then ( F ( E ) , dl) is necessarily nonseparable. The Hausdorff metric can be successfully applied in establishing the classical result of Suslin which states the existence of analytic non-Bore1 subsets of uncountable Polish spaces (see, for instance, [65], [89] or [97]). For the sake of completeness, we shall give a short proof of this result, especially taking account of the fact that the proof is essentially based on Theorem 2 which is the main tool for constructing various Peano type functions.
Theorem 4. Let E be an arbitrary uncountable Polish topological space (or, more generally, an uncountable Borel subset of a Polish space). Then there exists an analytic subset of E which is not BoreL Proof. Since all uncountable Borel subsets of Polish spaces are Borel isomorphic, it suffices to show that, in the standard Cantor space C = 2W, there exists an analytic subset which is not Borel. In order to do this, let us take the product space
and observe that it is compact. Denote by F ( W ) the family of all nonempty closed subsets of W and equip F ( W ) with the standard Hausdorff metric (or, equivalently, with the Vietoris topology which is met rizable by this metric). It can easily be checked that, in such a way, F ( W ) becomes a
CANTOR A N D PEANO T Y P E FUNCTIONS
47
compact metric space. According to Theorem 2, there exists a surjective continuous mapping h : 2W-, F ( W ) . Further, let Z denote the set of all irrational points of the segment [ O , l ] (we recall that Z is homeomorphic to the canonical Baire space w W where w is equipped with the discrete topology). Now, we define a set-valued mapping
by the following formula:
It is clear that (2Wx 2 ) n h ( t ) ranges over the family of all closed subsets of the space 2Wx Z as t ranges over 2W.By starting with this fact, it is not difficult to see that ran(@) = A(2W) where A(2W)denotes the family of all analytic subsets of 2W.Let us put
and establish that X is not analytic in 2W.Suppose, for a while, that X is analytic. Then, for some to E 2W,we must have the equality X = @(to). But, according to the definition of .X, we get
which obviously yields a contradiction. Consequently, X is not an analytic subset of the Cantor space. On the other hand, let us verify that the set
is an analytic subset of the Cantor space. Indeed, we may write
where the set D C_ W is defined by the formula
Since our mapping h is continuous, the set
48
C H A PT E R 1
is closed in W and, therefore, is a Gpsubset of W. Also, the set
is a G6-subset of W. Hence the intersection Dl n Dl1 = D is a Gpsubset of W , too, and prl(D) is an analytic set in the Cantor space 2W.Finally, we may infer that prl(D) is not Borel because, M has been shown above, the set X = 2W\ prl(D) is not analytic. This ends the proof of Theorem 4.
Example 1. In connection with Theorem 4, it is reasonable to point out that, in classical mathematical analysis, there are many interesting concrete sets of functions, analytic but not Borel (in an appropriate Polish space). For instance, Mauldin established in [I031 that the set of all continuous nowhere differentiable real-valued functions defined on the unit segment [0, 11 is an analytic but not Borel subset of the separable Banach space C[O, 11. Many other such sets can be constructed in the theory of trigonometric series (for more information, see especially [65] and the references therein). Now, we are going to prove the following statement due to Rogers [123].
Theorem 5. There exists a Gs-subset B o f R such that its difference set B - B is not Borel. Proof. We begin with some simple observations., First of all, let us introduce two subsets C1and Cz of the unit segment [0, 11. Namely, the set C1 consists of all real numbers x with decimal expansions
where t , = 0 or t , = 1 for every natural index n; analogously, the set Cz consists of all real numbers y with decimal expansions
where t , = 0 or t , = 2 for every natural index n. Obviously, C1and C2are uncountable closed subsets of the segment [O, 11. Consequently, applying Theorem 4 proved above, we may choose an analytic subset A1 of Cl which is not Borel. Also, we can represent the set A1 M the projection (on the x-axis) of a certain Ga-subset Z1 of the Cartesian product C1x C2C R2.
49
CANTOR AND PEANO T Y P E FUNCTIONS
Now, let us consider a mapping q5 : R2
--+
R defined by the formula
This mapping is continuous. Moreover, by taking account of the definitions of C1 and Cz, it can easily be checked that the restriction of q5 to the product set C1 x C2 is a homeomorphism between C1 x C2 and q5(C1 x Cz). Hence Zo =q5(Z1) = { X t Y : ( x , Y ) € 2 1 ) is a G6-subset of the compact set $(C1 x C2). Finally, we infer that Zo is a Gpsubset of the real line. Now, let us put
Evidently, B is a Gd-subset of R. We are going to show that the difference set of B is not Borel in R. Let us denote this difference set by
+
It is enough to establish that the set D n (C1 3) is not Borel. First, let us note that the following inclusions are fulfilled:
Further, it is not difficult to check that each point of the set D n (C1 is of the form u - v where
But the relation u E Zo is true if and only if u
+ 3)
= x + y where
Consequently, D fl (C1 + 3) is the set of points of the form x + y - v where
X E C ~y, ~ C 2 ,v + 3 E C z l ( x , y ) € Z ~ . Now, let us remark that if the points x and y are fixed and (x, y) E then there exists only one point v satisfying the relations
21,
C H AP T ER 1
50
Namely, such a point is v following set: {a: In other words,
= y - 3. Hence D n (4+ 3) coincides with the
+3
: (3y)((a:, Y)
D n (C1 + 3 )
E 21)).
= A1 + 3 .
Since A1 is not Borel, the set D n (C1+ 3) is not Borel, either. This also shows us that the difference set D is not Borel in R. Theorem 5 has thus been proved. Furthermore, putting we get two Gpsubsets X and Y of R for which the vector sum X + Y is not Borel in R. We now wish to present another application of Theorem 2. Namely, we are going to prove the classical Banach-Mazur theorem on the universality of the space C[O, 11for the class of all separable metric spaces. Let M be a class of metric spaces and let X be some space from this class. We shall say that X is universal for M if, for any space Y belonging to M , there exists an isometric embedding of Y into X. In other words, X is universal for M if all spaces from M can be realized as isometric copies of corresponding subsets of X . Let M, denote the class of all separable metric spaces. The first example of a space universal for M, was constructed by Urysohn. Later, Banach and Mazur discovered that the classical function space C[O, 11 is universal for M,, too. In order to establish this fact, we need two lemmas.
Lemma 1. For a n y m e t r i c space ( X Id ) , there eaists a B a n a c h space E such that: 1) X can be ~someln'callyembedded i n E; 2) the weaght o f E is equal t o ihe weight of X ( i n particular, if X i s separable, t h e n E is separable, too).
Proof. We may assume, without loss of generality, that X # 0. Let us fix a point t E X. Further, for any x E X, let us define a function
by the formula
C A N T O R A N D PEANO TYPE FUNCTIONS
Obviously, fx is continuous, and the relation
indicates directly that f, is bounded. Now, let the symbol C b ( X ) denote the Banach space (with respect to the standard sup-norm) of all real-valued bounded continuous functions defined on X . We introduce a mapping
by the following formula:
Let us check that 4 is an isometric embedding of X into C b ( X ) , Indeed, for any two elements x E X and x' E X , we may write
But it can easily be observed that
Consequently, we get
which shows that $ is an isometric embedding. Let us define E as the closed vector subspace of C b ( X )generated by the set $ ( X ) . Then E is obviously a Banach space whose weight is equal to the weight of $ ( X ) or, equivalently, to the weight of X . We thus conclude that E is the required Banach space. Before the formulation of the next lemma, we need some auxiliary notions from the theory of topological vector spaces (below, we restrict ourselves to the class of topological vector spaces over the field R). Let E and E' be any two vector spaces over R. We say that these spaces are in a duality if a bilinear function
is given such that: a) for each x # 0 from E, the partial linear functional
CHAPTER 1
52
is not identically equal to zero; b) for each y # 0 from E', the partial linear functional
is not identically equal to zero. In such a situation, it is usually said that establishes a duality between the given spaces E and E'. In particular, let E be an arbitrary Banach space (or, more generally, normed vector space). As usual, we denote by E* the vector space of all continuous linear functionals on E. Evidently, a bilinear function
defined by the formula @(u,x) = u(x)
(u E
E*,x
E E)
establishes a duality between E* and E, Let X be a subset of E. We equip E* with the weakest topology r ( E * , X) for which all linear functionals from the family (a(.,x ) ) , ~are ~ continuous. Clearly, the pair ( E * , r ( E * , X)) is a topological vector space.
Exercise 18. Let X be a dense subset of a normed vector space E. Denote by B the closed unit ball in E*, i.e, put
Verify that the topologies on B induced by r ( E * , X ) and r ( E * , E ) are identical.
Lemma 2. Let E be a separable n o r m e d vecior space a n d let B denote t h e closed unat ball in E*. T h e n B i s a compact m e t r i c space w i t h respect t o t h e topology anduced b y r ( E * ,E ) . Proof. According to Exercise 18, the topology r ( E * , E ) restricted to B coincides with the topology
restricted t o B , where {xn : n E w ) is a countable dense subset of E. Let us define a mapping h : B-+RW
CANTOR AND PEANO T Y P E FUNCTIONS
by the formula (26 E B ) . h(u) = (u(x,)),,w Actually, h maps B into the product space
which obviously is compact and metrizable. Now, by starting with the definition of r ( E * ,{x, : n E w ) ) , it is not hard to demonstrate that B is homeomorphic to some closed subset of the above-mentioned product space and, consequently, B is compact and metrizable as well. Lemma 2 has thus been proved. We are ready to formulate and prove the following classical result of Banach and Mazur.
Theorem 6. The space C[O, 11 is universal for the class M,. Proof. Let X be an arbitrary separable metric space. According to Lemma 1, there exists a separable Banach space E containing an isometric copy of X . Hence it suffices to show that E can be isometrically embedded in C[0,1]. Let us denote by B the closed unit ball in E*. Since E is separable, the ball B equipped with the topology induced by r ( E * , E) is compact and metrizable (see Lemma 2). According to Theorem 2, there exists a continuous surjection g from the Cantor space C C [O,1] onto B . It is easy to see that g can be extended to a continuous mapping acting from [ O , l ] onto B (cf. the proof of Theorem 3). For the sake of simplicity, the extended mapping will be denoted by the same symbol g. Now, let us take an arbitrary element x from E and define a function
by the formula fx(t) = g ( t ) ( ~ ) (t E IO,lI). Since g is continuous, fx is continuous, too. Moreover, since g(t) E B for eacht E [O,l], we get
i.e, fx is also bounded. If y is another element from E, then, for any t E [O,l], we may write
54
CHAPTER 1
On the other hand, a simple consequence of the Hahn-Banach theorem says that if x # y, then there exists a continuous linear functional satisfying the relations In particular, u E B and, since g is a surjection from [0, 11 onto B, there exists a point t o E [O,1] such that 21 = g(to), Then, for the point to, we have IfX(t0) - fy(to)l = lu(x - Y)l = Ilx - YII, which shows us that llfx - f Y l l = IIx - vll and, consequently, the mapping X
+
fx
(X
E E, fx
E C[O, 11)
is an isometric embedding. This completes the proof of Theorem 6. Remark 2. According to the preceding result, the space C[0,1] is universal for the class Ma. At the same time, it is obvious that C[O, 11 cannot be universal for the class M(c) consisting of all those metric spaces whose cardinalities are less than or equal to the cardinality of the continuum (clearly, we have the proper inclusion Ma c M(c)). The problem of the existence of a metric space X E M(c) universal for M(c) was investigated, with related problems for other infinite cardinals, by Sierpiriski (see 11351 and [137]).
Evidently, an analogous question of the existence of universal objects can be posed for various mathematical structures: for algebraic structures (e.g. groups), binary relations, topological structures and so on. There are some important results in this direction. For instance, let us recall the well-known theorem of Cantor stating that the set Q equipped with its standard order is universal for the class of all countable linearly ordered sets, i.e. every countable linearly ordered set can be isomorphically embedded in Q. The following example illustrates the situation for linearly ordered sets whose cardinalities are less than or equal to c. Example 2. Suppose that the Continuum Hypothesis holds. Then there exists a linearly ordered set ( X , <) with card(X) = c such that, for any linearly ordered set (Y, 5 ) with card(Y) 5 c, there is a monomorphism from Y into X. For the proof, see e.g. [I341 where related results are also discussed.
2. Singular monotone functions
This chapter is devoted to some elementary properties of monotone functions acting from R into R, and to some widely known examples of strange monotone functions. Let
be a function. We recall that f is said to be increasing (respectively, strictly increasing) if, for any two points x E R and y E R, the relation x y (respectively, x < y) implies the relation f (x) f(y) (respectively, f (x) < f (y)). Analogously the notion of a decreasing (respectively, strictly decreasing) function can be introduced. It is easy to see that f is increasing (strictly increasing) if and only if - f is decreasing (strictly decreasing). A function from R into R is said to be monotone (respectively, strictly monotone) if it is either increasing or decreasing (respectively, either strictly increasing or strictly decreasing). We shall consider below only increasing functions (this, of course, does not restrict the generality of our considerations).
<
<
If f is an increasing function and x E R, then there exist limits
and we have the inequalities
Obviously f is continuous a t x if and only if
Thus, x is a discontinuity point o f f if and only if
56
C HAP T ER 2
More generally, let g : R -+ R be an arbitrary function and let x E R. We say that x is a simple discontinuity point for g if there exist g(x-) and 9(x+), but s(x-1 # 9(x> v s(x+) # 9(x). Evidently, if g is a monotone function, then all discontinuity points for g are its simple discontinuity points. We have the following useful result.
Theorem 1. Let f be an arbitrary function actang from R into R . Then the set of all simple dascontanuity points for f is at most countable.
Proof. Denote by E the set of all simple discontinuity points for f . Denote also: El = {a: E E : f(x-) < f(x+)),
Then we can write
E = ElUE2UEsUE4, ant it suffices to demonstrate that each set Es (i = 1,2,3,4)is at most countable. Since the argument is similar for all Ea,we shall only show that
For this purpose, let us define a mapping
such that
The existence of such a mapping is obvious. Now, it is not hard to show that this mapping is injective. Indeed, suppose to the contrary that, for some distinct points xl and xz from R , we have the equality
57
SINGULAR MONOTONE FUNCTIONS
Without loss of generality we may assume that xl
< xs.
Choose any point
t from Isl,x2[. Then we must have simultaneously
f ( t ) < ~ ( 5 2=) P ( x ~ ) , f ( t ) > p(x1) = P ( ~ 2 ) l which is impossible. The contradiction obtained establishes that the mapping x (P(x),q(x), ~ ( x ) ) (x E E l )
-
is injective, which, obviously, implies the relation
Theorem 1 has thus been proved. Exercise 1. Recall that a function f : R -, R possesses the Darboux property if, for each subinterval [a, b] of R, the range o f f contains the segment with the end-points f (a) and f ( b ) . Demonstrate that a function with the Darboux property has no simple discontinuity points. In particular, infer from this fact that i f f is the derivative of some function acting from R into R, then f has no simple discontinuity points. As a trivial consequence ofTheorem 1, we obtain that, for any monotone function f acting from R into R, the set of all discontinuity points o f f is a t most countable. Exercise 2. Let E = {xn : n E N ) be an arbitrary countable subset of R and let {rn : n E N) be a countable family of strictly positive real numbers, such that Tn
< +m.
neN For any x E R, let us put
where N(x) = {n E N : xn < x). In this way a certain function f from R into R is defined. Show that: a) f is increasing; b) f is continuous a t each point from R \ E; c) for any natural index n, we have
58
C H A PTER 2
in particular, f is discontinuous at each point of the given set E. Deduce from this result that if E is everywhere dense in R (for example, if E = Q), then the function f constructed above has an everywhere dense set of its discontinuity points. We now are going to present the classical Lebesgue theorem concerning the differentiation of monotone functions. For this purpose, we need three simple lemmas (cf. [ill]). First, let us recall the notion of a derived number for functions acting from R into R. Suppose that [a, b] is a segment of R and that f : [a, b] --+ R is a function. Let x E [a, b]. We say that t E is a derived number (or a Dini derived number) of f at x if there exists a sequence {xn : n E N ) of points from [a, b] tending to x, such that
In this case we shall write t = fb(x). One more remark. For any two real numbers t l and t z , it will be convenient to denote below by the symbol [ t l , t z ]the segment of R with the end-points t l and t z . Thus, we do not assume in this notation that t l < ts. Lemma 1. Let f : [a, b] --+ R be a strictly increasing function, let q be a positive real number and let X be a subset of [a, b] such that, for any point x E X , there eaists at lead one derived number fb(x) 5 q. Then we
have ihe inequality A*(f (X))
< qA*(X)1
where A is the standard Lebesgue measure on R. Proof. Fix an arbitrary that
X
E
> 0 and take an open subset G of [a, b] such
c G,
X(G)
< A*(X) + E .
Consider the family of segments
( f (x + h) - f (x))lh I q + €1. Clearly, this family forms a Vitali covering for the set f ( X ) . Consequently, there exists a disjoint countable family
such that
59
SINGULAR MONOTONE FUNCTIONS
Note that, since our function f is strictly increasing, the countable family of segments {[xn, xn hn] : n E N )
+
is disjoint, too, and the union of this family is contained in G. So we can write
Taking account of the fact that E > 0 was chosen arbitrarily, we conclude that A*(f (XI) qA*(X),
<
and the proof of Lemma 1 is complete. Lemma 2. Let f : [a, b] + R be a strictly increasing function, lei q be a positive real number and let X be a subset of [ a ,b] such that, for any point x E X , there eaists at least one derived number fb(x) 2 q. Then we have the inequality X*(f (XI) L qA"X).
Proof. As we know, the set of all discontinuity points for f is at most countable. Taking this fact into account, we may assume without loss of generality that f is continuous at each point belonging to the given set X. Now, if q = 0, then there is nothing to prove. So let us suppose that q > 0. Pick an arbitrary E > 0 for which q - E > 0. There exists an open set G C R such that
Consider the family of segments
Obviously, this family forms a Vitali covering for the set X . Consequently, there exists a disjoint countable family
CHAPTER
2
for which we have
A(X\U{[x,,x,
+ h,]
: n E N ) ) = 0.
Again, since our f is strictly increasing, the countable family of segments
will be disjoint, too, and the union of this family is contained in G. Hence we may write
Taking account of the fact that c > 0 is arbitrarily small, we come to the desired inequality qA*(X>I A*(f ( X ) ) . Lemma 2 has thus been proved.
Lemma 3. Let f : [a,b] -+ R be a strictly increasing function and let X = { x E [a,b] : there exist t w o distinct derived numbers o f f at x). Then X is a set of A-measure zero.
Proof. For any two rational numbers p and q satisfying the inequalities
let us denote
X P , q= { x E [a,b] : there exists a derived number of f at x less than p, and there exists a derived number of f at x greater than q). Clearly, we have
So it suffices to show that each set X p , , is of A-measure zero. Indeed, according to Lemma 1, we may write
SINGULAR MONOTONE FUNCTIONS
At the same time, according to Lemma 2, we have
These two inequalities yield
or, equivalently, 0 I (P- q)X*(Xp,q).
Since p - q lemma.
< 0,
we must have X*(Xplq)= 0. This ends the proof of the
We are now ready to present the classical Lebesgue theorem on differentiability (almost everywhere) of monotone functions.
Theorem 2. Lei f : [a, b] 4 R be a monotone function. Then f a's differentiable at alnaost all (with respect to A) points of [a, b ] . Proof. Obviously, we may suppose that f is increasing. Moreover, since the set of all differentiability points for f coincides with the set of all differentiability points for fi, where
and fi is strictly increasing, we may assume without loss of generality that our original function f is also strictly increasing. Now, in view of Lemma 3, it suffices only to demonstrate that the set
X = { x E [a,b] : for each n E N , there exists a derived number f b ( x ) 2 n) is of X-measure zero. But this follows directly from Lemma 2, because, in conformity with this lemma, we may write
for every natural number n, which immediately yields the required equality A*(X) = 0. This completes the proof of Theorem 2. It follows a t once from this theorem that a nowhere differentiable realvalued function f defined on a segment [a, b] is simultaneously nowhere
CHAPTER 2
62
monotone on [a, b], i.e, there does not exist a nondegenerate subinterval of [a, b] on which f is monotone. Exercise 3. Show that there exist continuous functions
which cannot be represented in the form
where gl and g2 are monotone functions acting from R into R. Exercise 4. Let f : [a, b] Show that
+R
be an increasing continuous function.
Give an example where this inequality is strict. In addition, demonstrate that if
then the function f is absolutely continuous on the whole segment [a, b ] . E x e r c i s e 5. Let A denote, as usual, the standard Lebesgue measure on
R and let X be a Lebesgue measure zero subset of R. Then there exists a sequence {Un : n E N ) of open subsets of R, such that X
A(Un) < 1/2n
U,,
( n E N).
For any n E N , let us define
Then fn is increasing, continuous and 0
5 f n ( ~ )< 112"
for all x E R. Further, define
Show that the function f x is increasing, continuous and, for any point x E X , the equality
SINGULAR MONOTONE FUNCTIONS
63
holds true. For our further considerations, we need the following useful result due to Fubini.
Theorem 3. Let {Fn : n E N ) be a sequence of increasing functions given on a segment [a, b], such that: a) for each x E [a, b] and for any natural n, we have F n ( x ) 2 0; b) for each x E [a, b], we have
e on [a, b]) points Then, for almost all (with respect to the ~ e b e s ~ umeasure x E [a;b], the equality
as satisfied.
Proof. Clearly, we may write
for all those points x E [a, b] where the derivatives exist. In other words, FA is convergent almost everywhere on [a, b]. the series of functions CnEN Now, denote
and, for any natural k , choose an index n ( k ) such that
Since all Fn are increasing, we also have
for each x E [a, b]. This implies that the series
CHAPTER 2
64
converges uniformly on [a, b] to some increasing function. Consequently, the series (F(x) - sn(k)(x))l converges at almost all points x E [a, b]. From this fact we infer that
for almost all x E [a, b], i.e.
for almost all x E [a, b]. But this immediately yields that
almost everywhere on [a, b], The theorem has thus been proved. The next exercise provides an application of this theorem to the differentiation of an indefinite Lebesgue integral. Exercise 6. Let f be a positive lower semicontinuous function given on a segment [a,b]. We recall (see Chapter 0, Exercise 11) that f can be represented in the form
f =s u ~ n e ~ f n , where all functions fn are positive, too, and continuous. Derive from this fact that f can be also represented in the form
where all functions gn are positive and continuous. Let now f be a positive, Lebesgue integrable, lower semicontinuous function on [a, b] and let
Show, by applying the fact formulated above and Theorem 3, that
for almost all (with respect to the Lebesgue measure) points x E [a, b].
65
S I N G U L A R M O N O T O N E F UN C TI O N S
Let g be a positive, Lebesgue integrable function on [a,b]. Show that, for each E > 0, there exists a lower semicontinuous function f on [a, b], such that g 5 f and
Deduce from this fact that there exists a sequence { f n : n E N ) of Lebesgue integrable lower semicontinuous functions, such that: a ) fn+l S f n for any n E N; b) g 5 f n for any n E N ; C ) limn,+, fn(z) = g(z) for almost all points z E [a, b]. In particular, we may write
almost everywhere on [a, b]. Observe that
Putting F(x) =
l X ( f 0-
g)(t)dt
(x E la,b1)1
and applying Theorem 3 again, demonstrate that
for almost all z E [a, b]. Finally, prove the Lebesgue theorem stating t h a t if h is an arbitrary real-valued Lebesgue integrable function on [a, b], then
for almost all z E [a, b] T h e exercise presented above shows us that the classical Lebesgue theorem concerning the differentiation of a function H, where
66
CHAPTER
2
can be logically deduced from Theorem 3. However, this approach has a weak side because it does not yield the description of the set of those points z E [a, b] at which H'(x) = h(x). We now turn our attention t o the construction of a strictly increasing function whose derivative vanishes almost everywhere. Such a construction is essentially based on Theorem 3. Let us recall that the first step of the construction of the Cantor set on R is that we remove from the segment [0, 11 the open interval ]1/3,2/3[. Let us define f(x)=O (~50)l
Now, suppose that on the n-th step of the construction we have already defined the function f for all those points which belong t o the union of the removed (at this and earlier steps) intervals. Obviously, we obtain a finite family {[a,, bi] : 1 5 i 5 m) of pairwise disjoint segments on [O, 11. It is easy t o check that m = 2", but we do not need this fact for our further purposes. Pick any segment [ad, bi] from the above-mentioned family. Taking into account the inductive assumption, we may put
for all points x E](2ai+ bi)/3, (2bi +aa)/3[. So we have defined our function f for all points belonging t o the union of all intervals removed at the ( n + l ) t h step. Continuing the process in this way, we will be able t o construct f on the set R \ C, where C denotes the Cantor set. From the definition of f immediately follows that f is increasing and continuous on its domain. Moreover, it is easily seen that f can be uniquely extended t o an increasing continuous function f : R+[O,l]. Since f is constant on each removed interval, we obviously have fl(x) = 0
(x E R \ C),
i.e, the derivative of f vanishes almost everywhere on R. Thus, we have shown that there exists a non-constant increasing bounded continuous function f from R into R, whose derivative is zero almost everywhere. Now, let p and q be any two points of R such that p < q. Since f is not constant, there are some points x and y from R such that f (x) < f (y).
67
SINGULAR MONOTONE FUNCTIONS
Evidently, x < y and there exists a homothety (or translation) h of the plane R=,for which
Let f+ denote the function from R into R, whose graph coincides with t h e image of the graph o f f with respect t o h. Then we may assert that f * is also a n increasing bounded continuous function, whose derivative vanishes almost everywhere, and f * ( p ) < f*(q). In virtue of the remarks made above, we can formulate and prove the following classical result concerning the existence of strictly increasing continuous singular functions.
Theorem 4. There exasts a function g :
R- R
satisfying these three conditions: 1) g is continuous and strictly increasing; 8) (Vz E R)(O I g(z) _< 1); 3) the derivative of g as zero almost everywhere on R.
Proof. Let {(p,,q,) : n E N ) denote the countable family of all pairs of rational numbers, such that p, < q,. According t o the argument presented above, for each natural index n, there is a function
such that: a) g, is continuous and increasing; b) 0 g,(x) _< 1/2"'t1 for all x E R; c) the inequality g,(p,) < g, (9,) holds true; d) t h e derivative of g, vanishes almost everywhere. It follows from b) t h a t the series CnEN gn is uniformly convergent. So we may consider the function
<
which is continuous and increasing because of a). Evidently,
CHAPTER 2
In accordance with c), we also have
which immediately implies that g is a strictly increasing function. Finally, taking into account condition d ) and applying Theorem 3, we conclude that the derivative of g equals zero almost everywhere on R. E x e r c i s e 7. Let f be any continuous increasing function from R into R, whose derivative vanishes almost everywhere on R. For each half-open subinterval [a, b[ of R, let us put
Show that p can be uniquely extended t o a a-finite Borel measure on R (denoted by the same symbol p ) which is diffused (i.e. vanishes on all oneelement subsets of R) and is singular with respect t o the standard Lebesgue measure A. T h e latter means that there exists a Borel subset X of R for which we have A(X) = 0, p ( R \ X ) = 0. Formulate t h e converse assertion and prove it by utilizing the Vitali covering theorem.
3. Everywhere differentiable nowhere monotone functions
As mentioned in the preceding chapter, if a function f : R-+R is nowhere differentiable, then f is nowhere monotone, i.e. there does not exist a nondegenerate subinterval of R on which f is monotone. This chapter is devoted t o some constructions of functions also acting from R into R, differentiable everywhere but nowhere monotone. The question of the existence of such functions is obviously typical for classical mathematical analysis. And it should be noticed that many mathematicians of the end of 19-th century and of the beginning of 20-th century tried to present various constructions of the above-mentioned functions. As a rule, their constructions were either incorrect or, at least, incomplete. As pointed out in [64], the first explicit construction of such a function was given by Kopcke in 1889. Another example was suggested by Pereno in 1897 (this example is presented in [58], pp. 412 - 421). In addition, Denjoy gave in his extensive work [35] a proof of the existence of an everywhere differentiable nowhere monotone function, as a consequence of his deep investigations concerning trigonometric series and their convergence. Afterwards, a number of distinct proofs of the existence of such functions were given by several authors (see, e.g., [93], [64], [50], [165]). We begin with the discussion of the construction presented in [64]. This construction is completely elementary and belongs to classical mathematical analysis. We need some easy auxiliary propositions. Lemma 1. Let r and s be two strictly positive real numbers. The following assertions hold: I ) if r > S, then ( r - s)/(r2 - s 2 ) < 2/r;
2) if r > 1 and s > 1, then
CHAPTER
3
Proof. Indeed, we have
Thus 1 ) is true. Further, it can easily be checked that the inequality of 2) is equivalent t o the inequality
which, obviously, is true under our assumptions r completes the proof of Lemma 1.
Lemma 2. Let
> 1 and s > 1. This
4 be ihe function fiom R into R defined by
and let a and b be a n y two distinct real numbers. T h e n we h a v e
Proof. Without loss of generality we may assume that a < 6. Only three cases are possible. 1. 0 5 a < b. In this case, taking into account 1 ) of Lemma 1 , we can write ( l / ( b - a ) ) J b Q ( x ) d x = 2 ( ( 1 + b)li2 - ( 1 a
+ a ) ' 1 2 ) / ( ( l + b) - ( I + a ) )
2. a < b _< 0. This case can be reduced to the previous one, because of the evenness of our function q5. 3 , a < 0 < b. In this case, taking into account 2 ) of Lemma 1 , we can write ( l / ( b- a ) )
q5(x)dX = 2 ( ( 1 + b)'I2 a
+ ( 1 - a)'12 - 2 ) / ( ( 1 + 6 ) + ( 1 - a ) - 2 )
< 4 m i n ( 4 ( a ) ,4 ( b ) ) . This ends the proof of Lemma 2.
EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS
71
Lemma 3. Let n > 0 be a natural number and let $ : R -+ R be any function of the form
where $ is the function from Lemma 2,
are strictly positive real numbers, and
tl,
..., in are
real numbers. Then
for all de'stinct reals a and b.
Proof. The assertion follows immediately from Lemma 2, by taking into account the fact that
for any d > 0 and t E R.
Lemma 4. Let ( $ J , ) , be ~ ~a sequence offunctions as in Lemma 3. For any z E R and for each n 2 1, let U S define
and suppose that, for some a E R, the series Denote z $ . ( a ) = s < +m.
-
$,(a) is convergent.
tall
Then we have: I ) the seraes F ( z ) = - Q n ( z ) converges uniformly on every bounded subinterval of R; 2) the functa'on F is diflerentiable at a and
In particular, af
C a(4= f ( 4 <
$03
72
CHAPTER
3
for each a E R, then the functaon F is differentiable everywhere on R and the equality F' = f holds true.
Proof. Take any b E R satisfying the relation b 2 la[, In view of Lemma 3, for all x E [-6, b] and for all n 2 1, we may write
4Ial$n(a)
+ 4 1 -~ al+n(a) < 12b+n(a).
This shows the uniform convergence of 9,(x) on the segment [-b,b]. Further, let e > 0 be given. Pick a natural number k > 0 such that
Since all functions 4, are continuous on the whole R (in particular, at a), there exists some 6 > 0 such that
whenever O
1Ln
Consequently, assuming that 0 < Ih( < 6 and applying Lemma 3 again, we get
We thus conclude that F'(a) = s , and the lemma is proved. Lemma 5. Let n > 0 be a natural numbep; let 11,..., In be pairwise disjoint nondegenemte segments on R and let tk denote the midpoint of Ik for each natural k E [l,n]. Fix any strictly positive real numbers
EVERYWHERE. DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS
73
Then there exists Q function $ as in Lemma 3, such that, for all natural k E [I, n], the following relations are fulfilled: 1) $(tk) > Y k ;
2) (Vz E I k ) ( $ b ) < Y k +&I; 3) (Vz € R \ (11 U U In))($(z)< 6)-
Proof. Let us denote
Then, for each natural k E [ I , n], define
where dk > 0 is chosen so large that
Finally, put
$=$I+
+$nu
Then, taking into account the fact that
it is easy to check that the function $ satisfies relations 1)
- 3).
Lemma 6. Let any two disjoint countable subsets
of R be ge'ven. Then there exasts
Q
function
such that: 1) F is dzffereniiable everywhere on R ; 2) 0 c FFl(z)5 1 for ail x E R; 3) F f ( t k )= 1 for each k E N \ ( 0 ) ; 4) F'(rk) c 1 for each k E N \ ( 0 ) .
Proof. We are going t o construct by recursion the sequence ($n)nl 1 of functions as in Lemma 3 with some additional properties. Namely, denoting
CHAPTER 3
74
we wish the following conditions t o be satisfied: ( 1 ) for any natural n 2 1 and for all natural k E [ I , n ] ,we have
(2) for any natural n 2 1 and for each x E R , we have
(3) for any natural n 2 1 and for all natural k E [ I , n],we have
Ili First choose a nondegenerate segment II with midpoint t l , such t h a t rl $ and apply Lemma 5 with
Evidently, we obtain and f l = such t h a t relations ( I ) , ( 2 ) and (3) are fulfilled for n = 1. Suppose now t h a t , for a natural n > 1 , we have already defined t h e functions $1, 1 $kt i4n-1
--
---
satisfying the corresponding analogues of (1) - (3) for n - 1. Pick disjoint nondegenerate segments I l l ... , In in such a way that: (a) tk is the midpoint of Ik for each natural k E [ I , n]; (b) Ik n { r l , ... , r n ) = 0 for each natural k E [ I , n]; (c) for any natural k E [ I , n] and for any x E I k , we have t h e inequality
where
+
6 = l/(n(n 1))
-
1/(2n - 2n).
We now can apply Lemma 5 with
and ylc = 1 - ( l / n )- fn-~(tlc) ( 1
I k 5 n).
Applying th above-mentioned lemma, we get the function $,. Clearly,
EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS
75
for all natural k E [I, n], so relation (3) holds true. Further, for any natural k E [I, n], we have
which shows that relation (1) holds true, too. Finally, in order to verify (2), fix any point x E R. If, for some natural k E: [I, n], the point x belongs to Ik,then we may write
1- l/n
+ l/(n(n + 1)) = 1 - l / ( n + 1)-
If r: does not belong to I1U
...
U In,then
Thus, relation (2) holds true, too. Proceeding in this manner, we are able to construct the required sequence ($n)nL1. Putting
and
we obtain a function F : R
-+
R.In view of Lemma 4, we also get
F1(x) = f (x)
(x E R).
Further, the definition of F immediately implies 0 < F1(x) _< 1
(x t. R),
Now, fix a natural k 2 1 and let n be a natural number strictly greater than k. Then Fi(rk) = fn-l(ri) $m(rk) <
+C
naln
This completes the proof of Lemma 6.
CHAPTER 3
Theorem 1. There exasts a function
such that: I ) H is differentiable everywhere on R; 2) HI is bounded on R; 3) H zs monotone on no subinterval of R.
Proof. Denote by {t, : n E N , n
{r, : n E N , n > 0)
> 0),
any two disjoint countable dense subsets of R. Using the result of t h e previous lemma, take any two everywhere differentiable functions
satisfying the relations: (a) 0 < F'(x) 1 and 0 < G1(x) 1 for all x E R; (b) F1(tfl) = 1 and F 1 (r n ) < 1 for each natural n 2 1; (c) G 1 (r n ) = 1 and G 1 (t n ) < 1 for each natural n 1. Now, define H=F-G.
<
<
>
Obviously, we have
Since t h e sets { t , : n E N , n > 0) and {r, : n E N , n > 0 ) are dense in R, we infer t h a t H cannot be monotone on any subinterval of R. Also, the relation 1 -1 < H (x) < 1 (x E R ) implies that H' is bounded, and the theorem has thus been proved. In fact, the preceding argument establishes the existence of many functions f : R+R which are everywhere differentiable, nowhere monotone and such t h a t f' is bounded. Let us mention some other interesting properties of any such a function f.
EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS
77
1. f has a point of a local maximum and a point of a local minimum in every nonempty open subinterval of R. Actually, for each nondegenerate segment [a, b] C R, we can find some points t l and t z satisfying the relations
Let us denote M = suPt,[t,,t,] f ( t ) . Then, for some T E [tl,tz],we must have f ( r ) = M , and it is clear that T must be in the interior of [tl,tz]. In other words, T is a point of a local maximum for f . Applying a similar argument, we can also find a point of a local minimum of f on the same nondegenerate segment [a, b].
.
2. Since f' is bounded, the function f satisfies the so-called Lipschitz condition, i.e, for some constant d 2 0, we have
Note that, in the latter relation, we may put
In particular, f is absolutely continuous. This also implies that f ' is Lebesgue integrable on each bounded subinterval of R.
3. The function f 1is not integrable in the Riemann sense on any nondegenerate segment [a, b] C R. To see this, suppose otherwise, i.e. suppose that f' is Riemann integrable on [a, b]. Then, according to a well-known theorem of mathematical analysis, f' must be continuous a t almost all (with respect t o the standard Lebesgue measure) points of [a, b] (see, e.g., [Ill]). Taking into account the fact that f ' changes its sign on each nonempty open subinterval of R , we infer that f ' must be zero at almost all points of [a, b]. Consequently, f must be constant on [a, b], which is impossible. The contradiction obtained yields the desired result. 4. Being a derivative, the function f' belongs to the first Baire class, i.e. it can be represented as a pointwise limit of a sequence of continuous functions. Hence, in virtue of the classical Baire theorem (see, e.g., [ l l I]), the set of all those points of R a t which f ' is continuous is residual (comeager), i.e, is the complement of a first category subset of R.
5 . Let us denote
CHAPTER 3
78
The sets X and Y are disjoint, Lebesgue measurable and have the property that, for each nondegenerate segment [a, b] C R, the relations X(X
n [a, b]) > 0,
X(Y
n [a, b]) > 0
are fulfilled (where X denotes, as usual, the Lebesgue measure on R ) . In order to demonstrate this fact, suppose, for example, that X(Y n [a, b]) = 0. Then we get f t ( t ) _> 0 for almost all points t E [a, b]. But this immediately implies that our function f , being of the form
is increasing on [a, b], which contradicts the definition o f f . E x e r c i s e 1. Give a direct construction of two disjoint Lebesgue measurable subsets X and Y of R, such that, for any nonempty open interval I E R,the inequalities
hold true. More generally, show that there exists a countable partition {X,, : n < w ) of R consisting of Lebesgue measurable sets and such that, for any nonempty open subinterval I of R and for any natural number n , the relation X(I n x,) > o is fulfilled. E x e r c i s e 2. Denote by E the family of all Lebesgue measurable subsets of the unit segment [0, I]. For any two sets X E E and Y E E, put
Identifying all those X and Y , for which d ( X ,Y ) ' = 0, we come t o the metric space ( E l d). Check that ( E , d ) is complete and separable, i.e, is a Polish space. Further, let E' be a subset of E consisting of all sets X E E such that X(X n I ) > 0, X(([O, 11 \ X ) n I ) > o
EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS
79
for each nondegenerate subinterval I of [O,l]. Show that E' is the complement of a first category subset of E . Hence, according to the Baire Category Theorem, E' # 0. Now, we are going t o present an essential generalization of Theorem 1 due t o Weil (see [165]). Namely, Weil gave a proof of the existence of everywhere differentiable nowhere monotone functions (with a bounded derivative) by using the Baire Category Theorem. We shall say that a function
is a derivative if there exists a t least one everywhere differentiable function
satisfying the relation (Vx E R)(F1(x) = f (x)). Let us consider the set D = {f : f i s a derivative a n d f i s bounded} Obviously, D is a vector space over d defined by the formula
R.We may equip this set with a metric
T h e latter metric produces the topology of uniform convergence. In view of the well-known theorem of analysis, a uniform limit of a sequence of bounded derivatives is a bounded derivative. This shows, in particular, that the pair ( D , d ) is a Banach space (it can easily be seen that it is nonseparable). Take any function f E D and consider the set f-l(O). We assert that this set is a Ga-subset of R. Indeed, we may write
where F : R -+ R is such that F' = f . This formula yields a t once the desired result. Let us put Do = {f E D : f-l(0) i s dense in R). We need the following simple fact.
80
C HAPT E R
3
L e m m a 7. The set Do is a closed vector svbspace of the space D. Consequenlly, Do as a Banach space, as well. P r o o f . First, we show t h a t D ois closed in D. Let { f k : k < w ) be a sequence of functions from Do,converging (in metric d) t o some function f E D. We put (k < w ) . z k = fi1(0) Then all the sets 21,are dense G6-subsets of R. Therefore, the set
is a dense Ga-subset of R, too. Obviously,
Thus we obtain f E Do. Now, let us demonstrate that Do is a vector subspace of D. Clearly, if f E Do and t E R, then t f E Do.Further, take any two functions g E Do and h E D oand consider the sets
Then the set Zg n Zh is a dense Ga-subset of R, and it is evident that
where
+
Zgth = ( g h)-'(o), This shows that Dois a vector space. Lemma 7 has thus been proved. Notice now that the space Dois nontrivial, i.e, contains nonzero functions. For instance, this fact follows directly from Theorem 1. But it can also easily be proved by another argument. E x e r c i s e 3. Give a direct proof (i.e, without the aid of Theorem 1) that Do contains nonzero elements.
Theorem 2. Let us denote
E = {f E Do
:
there i s a nondegenerate subinterval of R
on which f preserves its sign). Then the set E is offirst category in the space Do.
EVERYWHERE DIFFERENTIABLE NOWHERE MONOTONE FUNCTIONS
81
P r o o f . Let {I, : n E N ) be the family of all subintervals of R with rational end-points. For each n E N , put
Bn ={f € D O : ( V z E I n ) ( f ( x ) S O ) ) . Clearly, we have
= UnEN(An UBn), so it suffices t o demonstrate that each of the sets A, and B, is closed and nowhere dense. We shall establish this fact only for A, (for B,, the argument is analogous). T h e closedness of A, is trivial. In order t o prove that A, contains no ball in Do, take any f E Do and fix an arbitrary E > 0. Since f E D o , there exists a point x E In such that f(x) = 0. Now, by starting with the existence of a nonzero bounded derivative belonging t o D o , it is easy t o show that there is a function h E D o for which
Let us define g=f+h. Then the function g belongs t o the ball of Do with centre f and radius At the same time, g does not belong t o A, because
E.
This establishes that A, is nowhere dense in D O , and the theorem has thus been proved. In the next chapter, we shall consider one more proof of the existence of everywhere differentiable nowhere monotone functions, by applying some properties of the so-called density topology on R. E x e r c i s e 4. Let E be an arbitrary topological space. We recall that a family N of subsets of E is a net in E if each open subset of E can be represented as the union of some subfamily of N (the concept of a net, for topological spaces, was introduced by Archangelski; obviously, it generalizes the concept of a base of a topological space). We denote by the symbol nw(E) the smallest cardinality of a net in E . Let now f : E+R be a function. We put:
82
CHAPTER
3
lmaxv(f) = the set of all those t E R for which there exists a nonempty open subset U of E such that t = s u p ( f ( U ) )and, in addition, there is a point e E U such that f ( e ) = s u p ( f ( U ) ) ; lminv(f) = the set of all those t E R for which there exists a nonempty open subset V of E such that t = i n f ( f ( V ) )and, in addition, there is a point e E V such that f ( e ) = in f ( f ( V ) ) . Check that card(lmaxv(f)) nw(E) w,
<
+
In particular, if E possesses a countable net, then the above-mentioned subsets of R are a t most countable. Denote also: slmax(f) = the set of all points e E E having the property that there exists a neighbourhood U ( e ) such that f ( e ) > f ( x ) for each x E U ( e ) ; slmin(f) = the set of all points e E E having the property that there exists a neighbourhood V ( e )such that f ( e ) < f ( x ) for each z E V ( e ) . Check that card(slmax(f)) nw(E)+ w,
<
In particular, if E possesses a countable net, then the sets slmax(f) and slmin(f) are at most countable. Finally, for E = R, give an example of a continuous function f for which the latter two sets are everywhere dense in E .
4. Nowhere approximately differentiable functions
The first example of a nowhere approximately differentiable function from the space C[O, 11is due to ~ a r n i k(see [60]). Moreover, he showed that such functions are typical, i.e. they constitute a set whose complement is of first category in C[O, 11. In this chapter we present one precise construction of a function acting from R into R, which is nowhere approximately differentiable. This construction is due to Mali [loll (cf. also [29] and [30]). It is not difficult and, a t the same time, is rather vivid from the geometrical point of view. We begin with some preliminary notions and facts. Let X denote the standard Lebesgue measure on R and let X be a Xmeasurable subset of R. We recall that a point x E R is said to be a density point for (of) X if limh,o, h>o X(X n [x - h, x h])/2h = 1. According to the classical Lebesgue theorem (see, e.g., Chapter 0), almost all points of X are its density points.
+
Exercise 1. Let (tn)nEN be a sequence of strictly positive real numbers, such that
Let X be a Lebesgue measurable subset of R and let x E R. Prove that the following two assertions are equivalent: 1) z is a density point of X ; X(X n[X - t n lx tn])/2tn = 1. 2) limn,+,
+
Exercise 2. Let X be a Lebesgue measurable subset of R and let
x E R. Show that the following two assertions are equivalent: 1) x is a density point of X ; 2) limh,o+, k + ~ + X(X n [x- h , z k])/(h k) = 1.
+
+
T h e notion of a density point turned out to be rather deep and fruitful not only for real analysis but also for general topology, probability theory
84
C HAPT E R 4
and some other domains of mathematics. For example, by use of this notion the important concept of the density topology on R was introduced and investigated by several authors (Pauc, Goffman, Waterman, Nishiura, Neugebauer, Tall and others). This topology was studied, with its further generalizations, from different points of view (see, e.g., [51], [117], [I211 and [158]). We shall deal with the density topology (and with some of its natural analogues) in our considerations below. Now, let f : R + R be a function and let x E R. We recall that f is said to be approximately continuous at x if there exists a A-measurable set X such that: 1) x is a density point of X ; 2) the function f l(X U {x)) is continuous at x. The next two exercises show that Lebesgue measurable functions can be described in terms of approximate continuity. Exercise 3. Let g : R -+ R be a function, let E be a fixed strictly positive real number and suppose that, for any A-measurable set X with A(X) > 0, there exists a A-measurable set Y E X with A(Y) > 0, such that
Demonstrate that there exists a A-measurable function h : R which we have (Vx E R)(lg(x) - h(x)l < 6 ) .
4
R for
Infer from this fact that if the given function g satisfies the above condition for any E > 0, then g is measurable in the Lebesgue sense. Exercise 4. Let f : R -+ R be a function. By applying the result of Exercise 3 and utilizing the classical Luzin theorem on the structure of Lebesgue measurable functions (see, e.g., [ I l l ] ) , show that the following two assertions are equivalent: a) the function f is measurable in the Lebesgue sense; b) for almost all (with respect to A) points x E R, the function f is approximately continuous a t x. Exercise 5. Let f : R + R be a locally bounded Lebesgue measurable function and let PZ
Prove that, for any point x E R at which the function f is approximately continuous, we have F'(z) = f (x). Check that the local boundedness of f is essential here.
N O W HERE A PPR O XIM A TELY D IFF ERE N TI A BLE F U N C T I O N S
85
Let now f : R -t R be a function and let x E R. We say t h a t f is approximately differentiable a t x if there exist a Lebesgue measurable set Y C R, for which x is a density point, and a limit
This limit is denoted by fAp(x) and is called an approximate derivative of f a t the point x.
Exercise 6. Demonstrate that if a function f : R -t R is approximately differentiable a t x E R, then f is also approximately continuous a t x. Exercise 7. Check that an approximate derivative of a function
a t a point x E R is uniquely determined, i.e. it does not depend on the choice of a Lebesgue measurable set Y for which x is a density point and for which the corresponding limit exists. Check also that the family of all functions from R into R approximately differentiable a t x forms a vector space over R.
Exercise 8.' If a function f : R -t R is differentiable (in the usual sense) a t a point x E R, then f is approximately differentiable a t x and fAp(x) = f ' ( x ) . Give an example showing t h a t the converse assertion is not true. For our purposes below, we need two simple auxiliary propositions.
Lemma 1. Let f : R -t R be a function, let z be a point of R and suppose that f is approximately differentiable at Z . Then, for any real number M1 > f A p ( 2 ) ,we have
Similarly, for any real number Mz < f A p ( 2 ) , we have
Proof. Since the argument in both cases is comletely analogous, we shall consider only the case of MI. There exists a A-measurable set X such t h a t x is a density point of X and
CHAPTER 4
Fix e
> 0 for which
+
fAp(x) e < M I . Then there exists a real 5 > 0 such t h a t , for any strictly positive h have
< 6, we
But, if 6 > 0 is sufficiently small, then
for all strictly positive h
< 6.
So we obtain the relation
and the lemma is proved. Actually, in our further considerations we need only the following auxiliary assertion which is an immediate consequence of Lemma 1.
L e m m a 2 . Let f : R -t R be a function, let x be a point of R and suppose that, for every strictly positive real number M , the relation
holds true. Then f is not approxamately differentiable at x .
In particular, suppose that two sequences {hk : k E N ) ,
{Mk : k E N )
of real numbers are given, satisfying the following conditions: 1) hk > 0 and Mk > 0 for all natural k ; 2.) limk,+,hk = 0 and limk,+,Mk = +m; 3 ) the lower limit
is strictly positive. Then we can assert that our function f is not approximately differentiable a t the point x . After these simple preliminary remarks, we are able t o begin the construction of a nowhere approximately differentiable function.
NOWHERE APPROXIMATELY DIFFERENTIABLE FUNCTIONS
First of all, let us put
f1(4/9) = 2/3, fi(519) fi
= 113, fi(619) = 213, fi(71Q) = 313,
($19) = 213, fi(919) = 3/3
and extend (uniquely) this partial function to a continuous function
+
in such a way that f l becomes linear on each segment [k/9, (k 1)/9] where k = 0 , 1 , ..., 8. We shall start with this function f l . In our further construction, we also need an analogous function g acting from the segment [O, 91 into the segment [O, 31, Namely, we put
+
Obviously, g is continuous and linear on each segment [k, k 11 where k = 0 , 1 , ...,8. Also, another function similar t o g will be useful in our construction. Namely, we denote by g* the function from [0,9] into [0,3], whose graph is symmetric with the graph of g , with respect t o the straight line {(x,y) E R x R : y = 3/21, In other words, we put g*(x) = 3 - g(x) for all x E [O, 91. Suppose now that, for a natural number n 1, the function
>
has already been constructed, such that: (a) fn is continuous; (b) for each segment of the form [k/gn, (k k E {O, 1,
s.,
+ l)/gn], where
gn - I},
the function fn is linear on it and the image of this segment with respect t o fn is some segment of the form b/3", ( j 1)/3"], where
+
j E {O, 1, ..,,3n - 1).
Let us construct a function
88
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4
For this purpose, it suffices t o define f n t l for any segment [k/gn, ( k + l ) / g n ] where k E { O , l , ...,gn - 1). Here only two cases are possible. 1, fn is increasing on [k/gn, (k l)/gn]. In this case, let us consider the following two sets of points of the plane:
+
((0, O), (0,3), (9,3), (9, O H ,
Since we have here the vertices of two rectangles, there exists a unique affine transformation h : R~-+ R2 satisfying the conditions
h(993) = ((k
+ l)/gnJfn((k + l)/gn)),
+ 1)/9", fn(k/gn))t o the segment [k/gn, (k + l)/gn] h(9,O) = ((k
Let the graph of the restriction of f n t l coincide with the image of the graph of g with respect to h. 2, fn is decreasing on [k/gn, (k l)/gn], In this case, let us consider the following two sets of points of the plane:
+
Here we also have the vertices of two rectangles, so there exists a unique affine transformation h* : R2--, R2 satisfying the relations
+
Let the graph of the restriction of f n + l t o the segment [k/gn, (k l)/gn] coincide with the image of the graph of g* with respect'to h*. T h e function f n t l has thus been constructed. From the above construction immediately follows that the corresponding analogues of the conditions
N O W HER E A P P R O X IM A TEL Y D IFFE R E N TI A BLE F U N C T I O N S
89
(a) and (b) hold true for f n + l , too. In other words, f n + l is continuous and, for each segment of the form [k/9"+',(k + 1)/9"+l], where
the function f n t l is linear on it and the image of this segment with respect to f n t l is some segment of the form b/3"t1, ( j 1)/3"t1], where
+
Moreover, our construction shows that
In addition, let [u, v] = [k/gn, (k
+ 1)/9"]
be an arbitrary segment on which fn is linear. Then it is not hard to check that fn+l([u, ( 2 ~ v)/31) = fn([u, ( 2 ~ v)/31),
+
+
+
+ + fn+l([(u + 2 ~ ) / 3 , v I )= fn([(u + 2 ~ ) / 3 , v I ) .
f n + l ( [ ( 2 ~ v)/3, ( 2 +~ ~1131)= f n ( [ ( 2 ~ v)/3, ( 2 ~ ~ ) / 3 1 ) , Proceeding in this way, we come to the sequence of functions
uniformly convergent to some continuous function f also acting from [O,1] into [0, 11. We assert that f is nowhere approximately differentiable on the segment [O,l]. In order to demonstrate this fact, let us take an arbitrary point x E [O,1] and fix a natural number n 2 1. Clearly, there exists a number k E {O,1, ...,gn - 1) such that
Therefore, we have
+
f n ( ~ E) Ij/3n, ( j 1)/3"1 for some number j E {0,1, ...,3" - 1). For the sake of simplicity, denote
From the remarks made above it immediately follows that, for all natural numbers m > n, we have f m ( 4 E b,ql
90
CHAP T ER
4
and, consequently, f (x) E b, q], too. Further, we may assume without loss of generality that fn is increasing on [u, v] (the case when f n is decreasing on [u, v] can be considered completely analogously). Suppose first that f (x) (p q)/2 and put
< +
Then, for each point y E Dl, we may write
Hence, we get ( f (y)
- f(x))/(y - 2) L ((2q + PI13 - (P + q)/2)l(v - 21) = (1/6)(3").
Suppose now that f (x) 2 (p
+ q)/2 and denote
DZ = [u, (2u
+ v)/3].
In this case, for any point y E Dz, we may write
Hence, we get ( f (XI - f(y))/(x
- y) L ((P + q)/2 - (
+
2 ~ q)/3)/(v
- u) = (1/6)(3").
Thus, in the both cases, we have X({y E [x
- l/gn, x + l/gn] \ {x)
:
+ l/gn] \ {x)
:
or, equivalently, X({y E [x - l / g n , x
The latter relation immediately yields that our function f is not approximately differentiable a t x (see Lemma 2 and the comments after this lemma). Remark 1. The function f constructed above has a number of other interesting properties (for more information concerning f , see [loll and
P O 1 1.
NOWHERE APPROXIMATELY DIFFERENTIABLE FUNCTIONS
91
Now, starting with an arbitrary continuous nowhere approximately differentiable function acting from [O, 11 into [0, 11, we can easily obtain an analogous function for R. We thus come t o the following classical result (first obtained by Jarnik in 1934).
Theorem 1. There exist continuous bounded functions acting from R into R, which are nowhere approximately digerentaable. R e m a r k 2. Actually, Jarnik proved that almost all (in the sense of the Baire category) functions from the Banach space C[O, 11 are nowhere approximately differentiable. Clearly, this result generalizes the corresponding result of Banach and Mazurkiewicz for the usual differentiability, Further investigations showed that analogous statements hold true for many kinds of generalized derivatives. The main tool for obtaining such statements is the notion of porosity of a subset X of R a t a given point 2 L. R. this interesting topic is out of the scope of our book, So we only refer the reader t o the fundamental paper [19] where several category analogues of Theorem 1 for generalized derivatives are discussed from this point of view.
ow ever,
'
In Chapter 11 of our book we give an application of a nowhere approximately differentiable function t o the question concerning some relationships between the sup-measurability and weak sup-measurability of functions acting from R x R into R. Since the concept of an approximate derivative relies essentially on the notion of a density point, it is reasonable to introduce here the so-called density topology on R and t o consider briefly some elementary properties of this topology. E x e r c i s e 9. For any Lebesgue measurable subset X of R, let us denote
d ( X ) = {x E R : x is a density point for X ) . Further, denote by Td the family of all those Lebesgue measurable sets Y E R, for which Y E d ( Y ) . Show that: 1) Td is a topology on R strictly extending the standard Euclidean topology of R ; 2) the topological space ( R , T a ) is a Baire space and satisfies the Suslin condition (i.e, each disjoint family of nonempty open sets in ( R , T d ) is a t most countable); 3) every first category set in ( R , T d ) is nowhere dense and closed (in particular, the family of all subsets of ( R , T d ) having the Baire property coincides with the Bore1 o-algebra of ( R , Td));
92
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4
4) a set X C R is Lebesgue measurable if and only if X has the Baire property in ( R , Td); 5) a set X C R is of Lebesgue measure zero if and only if X is a first category subset of ( R , Td); 6) the space ( R , Td) is not separable. T h e above-mentioned topology Td is usually called the density topology of R. E x e r c i s e 10. Let f : R -t R be a function and let x E R. Prove that the following two assertions are equivalent: 1) f is approximately continuous a t x ; 2) f regarded as a mapping from ( R , T d ) into R is continuous a t x. E x e r c i s e 11. By starting with the result of the previous exercise, show t h a t the topological space (R, Td) is connected. For this purpose, suppose t o the contrary that there exists a partition {A, B) of R into two nonempty sets A E Td and B E Td. Then define a function
by putting f (x) = 1 for all x E A, and f (x) = -1 for all x E B. Obviously, f is a bounded continuous mapping from ( R , Td) into R and hence, according t o Exercise 10, f is approximately continuous a t each point of R. Further, define F(z) = f(t)dt (x E R ) .
1'
By applying Exercise 5 of this chapter, demonstrate t h a t the function F is differentiable everywhere on R and
for each x E R. This yields a contradiction with the Darboux property of any derivative. One of the most interesting facts concerning the density topology states t h a t ( R , T d ) is a completely regular topological space (see, for instance, [I171 and [158]). This property of T d implies some nontrivial consequences in real analysis. To illustrate, we shall sketch here a proof of the existence of everywhere differentiable nowhere monotone functions by applying the above-mentioned fact (note that this approach is due t o Goffman [50]). E x e r c i s e 12. Consider any two disjoint countable sets
NOWHERE APPROXIMATELY DIFFERENTIABLE FUNCTIONS
93
each of which is everywhere dense in R. Taking into account Exercise 10 and the fact that (R,Td) is completely regular, we can find, for any n E N , a n approximately continuous function
satisfying the relations
Analogously, for any n E N , there exists an approximately continuous function gn : R -t [O, 11 such that 0
< gn(x) < 1
(X E
R),
Now, define a function h : R-tR by the formula
Check that: (1) h is bounded and approximately continuous; (2) h(a) > 0 for all a E A; (3) h(b) < 0 for all b E B. Let H denote an indefinite integral of h. Show that: (i) H is everywhere differentiable on R and H1(x) = h(x) for each x E R; (ii) H is nowhere monotone. We thus see that with the aid of the density topology on R it is possible t o give another proof of the existence of everywhere differentiable nowhere monotone functions acting from R into R (cf. the proof presented in Chapter 3). R e m a r k 3. T h e density topology on R can be regarded as a very particular case of the so-called von Neumann topology. Let (ElS, p ) be a space with a complete probability measure (or, more generally, with a complete
94
CHAPTER 4
nonzero o-finite measure). Then, in conformity with a deep theorem of von Neumann and Maharam (see, e.g., [loo], [117], [121], [159]), there exists a topology T = T ( p ) on E such that: 1) ( E , T ) is a Baire space satisfying the Suslin condition; 2) the family of all subsets of ( E , T ) having the Baire property coincides with the o-algebra S ; 3) a set X E E is of p-measure zero if and only if X is of first category in ( E , T ) . We say that T = T ( p ) is the von Neumann topology associated with the given measure space ( E , S, p). Note that T, in general, is not unique. This fact is not so surprising, because the proof of the existence of T is essentially based on the Axiom of Choice. There are many nontrivial applications of a von Neumann topology in various branches of contemporary mathematics (for instance, some applications to the general theory of stochastic processes can be found in [121]). R e m a r k 4. For the real line R, an interesting analogue of the density topology, formulated in terms of category and the Baire property, was introduced and considered by Wilczyriski in [166]. Wilczyriski's topology was then investigated by several authors. An extensive survey devoted t o the properties of this topology and t o functions continuous with respect to it is contained in [30] (see also the list of references presented there). R e m a r k 5. There are some invariant extensions of the Lebesgue measure A , for which an analogue of the classical Lebesgue theorem on density points does not hold. For example, there exist a measure p on R and a p-measurable set X C R , such that: 1) p is an extension of A; 2) p is invariant under the group of all isometric transformations of R ; 3) there is only one p-density point for X , i.e. there exists a unique point x E R for which we have
A more detailed account of the measure p and its other extraordinary properties can be found in [78].
5 . Blurnberg's theorem and Sierpihski-Zygmund function
In various questions of analysis, we need to consider some restrictions of a given function (e.g, acting from R into R ) , having nice additional properties which do not hold, in general, for the original function. In order to illustrate this, let us recall two widely known statements from t h e theory of real functions. T h e first of them is the classical theorem of Luzin concerning the structure of an arbitrary Lebesgue measurable function acting from R into R. Undoubtedly, this theorem plays the most fundamental role in the theory of real functions. Let X denote the standard Lebesgue measure on the real line. Let
be a function measurable in the Lebesgue sense. Then, according t o the Luzin theorem (see, e.g., [ I l l ] ) , there exists a sequence {D, : n E N ) of closed subsets of R, such that
and, for each n E N , the restricted function
is continuous. It immediately follows from this important statement t h a t , for any Lebesgue measurable function f : R + R, there exists a continuous function g : R 4 R such that
Indeed, it suffices to take a set Dn with X(D,) > 0 and then t o extend the function f 1 D, to a continuous function g acting from R into R (obviously, we are dealing here with a very special case of the classical Tietze-Urysohn
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theorem on the existence of a continuous extension of a continuous realvalued function defined on a closed subset of a normal topological space). In particular, we have the equality
where c denotes, as usual, the cardinality of t h e continuum. Also, we may formulate the corresponding analogue of the Luzin theorem for real-valued functions possessing the Baire property. This analogue is essentially due to Baire. Let
be a function having the Baire property. Then there exists a subset D of R such that: 1) the set R \ D is of first category; 2) the function f ID is continuous. In particular, since card(D) = c and cl(D) = R, we conclude that the restriction o f f t o some everywhere dense subset of R having the cardinality of the continuum turns out to be continuous. I t can easily be observed that the Luzin theorem and its analogue for t h e Baire property hold true in much more general situations. T h e following two exercises show this.
Exercise 1. Let E be a Hausdorff topological space, let p be a finite Radon measure on E and let pi denote the usual completion of p. Prove that, for any p'-measurable function
and for each E > 0, there exists a compact set Ii' E E for which these two relations are fulfilled: 1) p ( E \ I<) < &; 2) the restriction of f t o I
In other words, every Radon measure is perfect (let us recall that perfect probability measures play an essential role in the theory of random processes).
Exercise 2. Let E be an arbitrary topological space and let E' be a topological space with a countable base. Demonstrate that, for any mapping f : E-tE' having t h e Baire property, there exists a subset D of E satisfying the relations: 1) the set E \ D is of first category; 2) the function f ID is continuous. Infer from this fact that if E is a Baire space, then the restriction of f t o some everywhere dense subset of E is continuous. T h e preceding results point out the nice behaviour of measurable functions (respectively, of functions with the Baire property) on some subsets of R which are not small (in a certain sense). Namely, as mentioned above, an immediate consequence of the Luzin theorem is that, for any Lebesgue measurable function f acting from R into R, there exists a closed subset A of R with strictly positive measure, such that the restriction f lA is continuous. T h e corresponding analogue for the Baire property states that, for any function g acting from R into R and possessing the Baire property; there exists a comeager subset B of R such that the restriction glB is continuous, too. In this connection, a natural question arises: what can be said about an arbitrary function acting from R into R ? In other words, if an arbitrary function f : R-+R is given, is it true that f ID is continuous on some set D S R which is not small (in a certain sense)? T h e first topological property of D that may be considered in this respect is the density of D in R. I t turns out t h a t this property is completely sufficient. Namely, Blumberg showed in [I 11 that there always exists a dense subset D of R for which the restriction f ID is continuous. This chapter of our book is devoted t o the Blumberg theorem and t o some strange functions which naturally appear when one tries t o generalize this theorem in various directions. Such strange functions were first constructed by Sierpiliski and Zygmund (see [143]). They are extremely discontinuous (more precisely, the restrictions of such functions t o any subset of R having the cardinality of the continuum are always discontinuous). We begin with the following auxiliary notion.
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Let f be a function acting from R into R and let x be a point of R. We shall say that x is a pleasant point with respect t o f (or, briefly, f-pleasant point) if, for each real E > 0, there exists a neighbourhood V(x) = V ( X , E ) such t h a t the set {Y E R : lf(9) - f ( x ) l < E) is categorically dense in V(x) (i.e, the intersection of this set with any nonempty open interval contained in V(x) is of second category in t h a t interval). In accordance with the definition above, we shall say t h a t a point x E R is f -unpleasant if x is not f -pleasant. The following key lemma shows t h a t , for any function f : R 4 R, the set of all f-unpleasant points is small in the sense of the Baire category, Lemma 1. Let f be a function acting from R into R . Then the set of all f-unpleasant points is of first category in R. Consequently, the set of all f -pleasant points is comeager on every nonempty open subanterval of R.
Proof. Suppose otherwise, i.e. suppose t h a t the set
C = {x E R : x i s f
- unpleasant)
is not of first category. For each point x E C, there exists a strictly positive number E(X)such t h a t , for any neighbourhood V of x, the set
is not categorically dense in V. Let us pick two rational numbers r(x) and s(x) satisfying the inequalities
f (x) - & ( ~ ) / < 2 r(x) < f ( 4 < s(x) < f ( 4 + & ( ~ ) / 2 . Further, for any pair (r, s ) of rational numbers, let us p u t
Evidently, we have the equality
Since, by our assumption, C is not of first category, there exists a pair (ro,so) E Q x Q such that the set Cr,,8, is not of first category, either. Consequently, there exists a nonempty open interval ]a, b[ C R such t h a t
the set C, is categorically dense in ]a, b[. Choose any point xo from the For this point, we may write set ]a, b[ n C,,,.
Analogously, for each point y E C,,,,,
, we have
f (y) - ~ ( ~ 1 <1 2Yo < f (Y) < S o < f (9) + & ( ~ ) / 2 . Therefore, If(!/) - f(xo)l < so In other words, we get the inclusion
-Yo
< ~(xo).
which immediately implies that the set
is categorically dense in ]a, b[ contradicting the definition of &(x0). T h e contradiction obtained ends the proof of Lemma 1. We are now ready t o prove the classical Blumberg theorem.
Theorem 1. Let f be an arbitrary function acting from R into R.Then there exists an everywhere dense subset X of R such that the function f lX is continuous. Proof. Since R is homeomorphic t o the open unit interval 10, I[, it suffices t o establish the assertion of Theorem 1 for any function
f
: IO,l[ -+ 10, I[.
Let I'(f).denote the graph of f . Taking Lemma 1 into account, we can recursively construct two sequences {Zn : n E N ) ,
{D, : n E N ) ,
satisfying the following conditions: 1) for each natural n , the set Zn can be represented in the form
where I ( n ) is a countable set, the intervals of the family {]ai, bi[ : i E I ( n ) ) are pairwise disjoint, the set
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is categorically dense in ]O,1[ and, for any i E I ( n ) , the length of ]cc, da[ is strictly less than l / ( n + 1); 2) for each natural n , the set D, is a finite ( l / ( n 1))-net of 10,1[, i.e. for any point t E 10, I[, there exists a point d E Dn such t h a t
+
3) the sequence of sets {Z, : n E N ) is decreasing by inclusion; 4) the sequence of sets {D, : n E N ) is increasing by incluzion; 5) for each natural n , we have
and any point from D, is f-pleasant. We leave the details of this construction t o the reader, because they are not difficult. As soon as the above-mentioned sequences are defined, we put
D = U{D,
:
n E N).
Then condition 2) implies that the set D is everywhere dense in ]O,l[, and it can easily be checked, by using conditions I), 3), 4)) 5), t h a t the restriction of f t o D is continuous. T h e Blumberg theorem has thus been proved.
Remark 1. Some more stronger versions of the Blumberg theorem may be obtained by using additional set-theoretical hypotheses. For instance, in [5] the situation is discussed when Martin's Axiom (or certain of its consequences) is assumed. Actually, Blumberg's theorem was analyzed and generalized in many directions. Moreover, the concept of a Blumberg topological space was introduced and investigated. Here we only formulate the corresponding definition. Namely, we say t h a t a topological space E is a Blumberg space if, for any function
there exists an everywhere dense subset X of E such t h a t the restriction glX is continuous. T h e class of all Blumberg spaces turns out to be rather wide and possesses a number of interesting properties. Let us point out t h a t any Blumberg space must be a Baire one (the reader can easily verify this simple fact). A useful survey of results concerning the Blumberg theorem and its generalizations is presented in paper [16] (see also [15], [61], [130]). We thus see that any function (acting from R into R ) restricted t o an appropriate countable everywhere dense subset of R becomes continuous.
In this connection, the question arises whether that subset can be chosen t o be uncountable. A partial negative answer t o this question yields the classical Sierpiliski-Zygmund function constructed by them in [143], with the aid of an uncountable form of the Axiom of Choice. This function has the property that its restriction t o each subset of R of cardinality continuum is discontinuous. Consequently, if the Continuum Hypothesis holds, then the restriction of the same function t o any uncountable subset of R is discontinuous, too. We shall present here the construction of the Sierpiriski-Zygmund function. Moreover, we shall give a slightly more general construction of a Sierpiliski-Zygmund type function possessing some additional properties. For this purpose, we need several auxiliary notions and statements. Let E be a Hausdorff topological space. We recall that E is normal if, for any two disjoint closed sets X C E and Y E E, there exist two open sets U C E and V E, such t h a t
In other words, E is normal if and only if any two disjoint closed subsets of E can be separated by disjoint open subsets of E. T h e well-known Tietze-Urysohn theorem (see, e.g., [89]) states that if E is a normal space, X is a closed subset of E and
is a continuous function, then there always exists a continuous function
extending f . Furthermore, if, for the original function f , we have the relation r a n ( f ) C [a, bl C R, then t h e extended function f * may be chosen t o satisfy an analogous relation r a n ( f *) c [a, bl. In fact, the property of the extendability of continuous real-valued functions defined on closed subsets of E is equivalent to the normality of E. A normal topological space E is called perfectly normal if each closed set in E is a Ga-subset of E (or, equivalently, if each open set in E is an F,subset of E ) . T h e following exercise yields another definition of perfectly normal spaces.
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102
E x e r c i s e 3. Let E be a normal space. Show that these two assertions are equivalent: 1) E is perfectly normal; 2) for any closed subset X of E, there exists a continuous function
such t h a t fil({O})
= X.
The next exercise indicates another important property of perfectly normal spaces. E x e r c i s e 4. Let E be a perfectly normal space and let M be some class of subsets of E, satisfying the following conditions: a) the family of all closed subsets of E is contained in M ; b) if {X, : n E N ) is an increasing (by inclusion) sequence of sets belonging to M , then the set u{X, : n E N ) belongs t o M , too; c) if {Y, : n E N } is a decreasing (by inclusion) sequence of sets belonging to M , then t h e set n{Yn : n E N ) belongs t o M , too. Prove, by applying the method of transfinite induction, that B ( E ) C M , i.e, each Borel subset of E belongs to M . T h e result presented in Exercise 4 enables us t o define Borel subsets of a perfectly normal space E in the following manner. First, we put B2;(E) = the class of all closed subsets of E. Suppose now t h a t , for a nonzero ordinal I < w l , the classes
have already been defined. If I is an odd ordinal, then we put BT(E) = the class of all those sets which can be represented in the form u{X, : n E N ) where {X, : n E N ) is some increasing (with respect to inclusion) sequence of sets belonging to the class U{BT(E) : 5 < I}. If I is an even ordinal, then we put B;(E) = the class of all those sets which can be represented in the form n{Yn : n E N} where {Yn : n E N ) is some decreasing (with respect t o inclusion) sequence of sets belonging t o the class U{BT(E) : C < I}. Finally, we define
Then, in virtue of the result of Exercise 4, we may assert t h a t
For any set X E B ( E ) , we say that X is of order ( if
Let now E be an arbitrary topological space. We denote by t h e symbol B ( E , R) the family of all Bore1 mappings acting from E into R. Furthermore, for any ordinal number ( < wl, we define by transfinite recursion the class B a E ( E ,R) of functions also acting from E into R. First of all, we put
where C ( E , R) is the family of all continuous real-valued functions on E. Suppose t h a t , for a nonzero ordinal 6 < wl, the classes of functions
have already been defined. Let us denote by B a C ( E ,R) the class of all those functions f : E-tR which satisfy the following condition: there exists a sequence {fn
: n E N}
E U{Bac(El R) : C < E )
(certainly, depending on f ) for which we have
In other words, B a < ( E ,R) consists of all pointwise limits of sequences of functions belonging t o U{Ba((E, R) : 5 < (1. Continuing in this manner, we are able to define all the classes
Finally, we p u t
For each ordinal ( < wl, the family of functions
is usually called the Baire class of functions having order (. A real-valued function given on E is called measurable in the Baire sense if it belongs to B a ( E , R).
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E x e r c i s e 5. Let E be a topological space. Prove, by using the method of transfinite induction, that: 1) for any E < C < wl, the inclusion
is fulfilled; 2) B a ( E , R ) is a vector space over R ; 3) if f E B a ( E , R ) and g E B a ( E , R ) , then f . g E B a ( E , R ) ; 4) if f E B a ( E , R ) , then ) f ) E B a ( E , R); 5) i f f and g belong to B a ( E , R ) , then
6) if f and g belong t o B a ( E , R ) and g(x) # 0 for all x E E, then f l g also belongs t o B a ( E lR ) ; 7) if a sequence {fn : n E N ) E B a ( E , R ) is given, such t h a t there exists a pointwise limit
then f belongs to B a ( E , R ) ; 8) if g E B a ( E , R ) is such that r a n ( s > E l a , b[ for some open interval ]a, b [ , then, for any continuous function
4 : la, b[
+
R,
the function 4 o g belongs to B a ( E , R ) ; 9) each function belonging to B a ( E , R ) is a Bore1 mapping from E into R ; in other words, we have the inclusion Ba(E,R )
E B ( E , R).
In some cases, this inclusion can be proper. For example, let us equip the ordinal number wl with its order topology. Check that
E x e r c i s e 6. Let cP be a class of functions acting from R into R. We say t h a t a function h : RxR-tR
is universal for cP if, for any function 4 E cP, there exists a point y = y(4) of R such that 4(.> = h(x, Y) (x E R ) . By starting with the existence of continuous mappings of Peano type (see Chapter 1) and applying the method of transfinite induction, show that, for each ordinal t < wl , there exists a function
satisfying the following conditions: a) hF is a Borel mapping from R x R into R; b) he is universal for the class B a t ( R , R ) . Deduce from this fact that, for any ordinal < wl, the set
is not empty (i.e, there are Baire functions of order t ) . In particular, the family of the Baire classes
is strictly increasing by inclusion. This remarkable result was first obtained by Lebesgue (see, e.g., [ I l l ] or [89]). L e m m a 2. For any perfectly normal space E , the equality
is fulfilled. Consequently, this equality holds true for an arbitrary metric space E (in particular, for E = R).
Proof. Obviously, it suffices to demonstrate that every bounded realvalued Borel function on E belongs to the class Ba(E, R). Since any such function is uniformly approximable by linear combinations of characteristic functions of Borel subsets of E, it is enough to show that the characteristic function of each Borel subset of E belongs to B a ( E , R ) . Let X be an arbitrary closed subset of E. Taking account of the perfect normality of E and applying the Tietze-Urysohn extension theorem, we can easily verify that the characteristic function xx is a pointwise limit of a sequence of continuous functions on E whose ranges are contained in [O, 11. Let now be an ordinal from the interval ]O,wl[ and suppose that the assertion is true for the characteristic functions of all Borel subsets of E
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belonging to the Borel classes of order strictly less than (. Take any Borel set X E E of order c. According to the result of Exercise 4, we may write X = U{X, : n E N ) (or, respectively, X = n{X, : n E N ) ) where {X, : n E N ) is an increasing (respectively, a decreasing) sequence of Borel sets in E belonging to some Borel classes of lower orders. But, in the both cases, we have X X = limn++oo XX,. In conformity with our inductive assumption, all functions xx, belong to the Baire class B a ( E , R). Hence the function x x belongs to this class, too. This completes the proof of Lemma 2. Let X be a topological space, let Y be a metric space and let Z be a subset of X . Suppose that a function
is given, and let x be an arbitrary point from X . We denote
where B ( x ) denotes a local base of X at z (i.e. B(z) is a fundamental system of open neighbourhoods of z ) . The real number o s c j ( x ) is usually called the oscillation of a function f at a point x . It Can easily be seen that if x does not belong to the closure of Z , then o s c f (8) = 0. Also, for a given function
the following two assertions are equivalent: 1) f is continuous; 2 ) for each point z E X , we have o s c j ( x ) = 0. The next auxiliary statement is due to Lavrentieff (see [94]). It has many important applications in general topology and descriptive set theory (cf., for instance, [89]). L e m m a 3. Let X be a metric space, let Y be a complete metric space and let Z be a subset of X . Suppose that a continuous function
i s given. Then there exist a set Z* C X and a function
BLUMBERG'S
THEOREM A N D SIERPI~SKI-ZYGMUNDFUNCTION
107
satisfying these three relations: 1) z c z * ; 2) Z* is a G6-subset of X; 3) f* is a continuous extension o f f .
Proof. Let c l ( Z ) denote the closure of Z in X . We put
Since our original function f is continuous, we have
(Vz E Z ) (oscf( z ) = 0) Consequently, Z C Z * . Now, let z be an arbitrary point of Z * . Then there exists a sequence { z , : n E N ) C Z such that
Taking account of the equality oscf ( z ) = 0, we see that
is a Cauchy sequence in Y . But Y is complete, so the above-mentioned sequence converges to some point y E Y . In addition, it can easily be observed that y does not depend on the choice of { z n : n E N ) . So we may put f * ( z ) = y. In this way we get the mapping
which is continuous because, according to our definition,
(Vz E Z * ) ( O S C ~ .=( ~0).) Thus, it remains to demonstrate that Z* is a Gs-subset of X . Obviously, the equality Z* = n { V n : n E N ) fl c l ( Z ) holds, where, for each n E N , the set Vn is defined as follows:
Since all V , ( n E N ) are open in X and the closed set c l ( Z ) is a Ga-subset of X , we infer that Z* is a Gpsubset of X , too. This finishes the proof of Lemma 3.
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Exercise 7. Let X and Y be two complete metric spaces, let A be a subset of X and let B be a subset of Y. Suppose also that
is a homeomorphism between A and B . By starting with the result of Lemma 3, show that there exist two sets A* C X and B* C Y and a mapping f * : A* + B * , satisfying these four relations: 1) A C A h n d B E B * ; 2) A* is a Gpsubset of X and B* is a Ga-subset of Y ; 3) f * is a homeomorphism between A* and B * ; 4) f * is an extension of f . This classical result was obtained by Lavrentieff [94] and is known as the Lavrentieff theorem on extensions of homeomorphisms. It found numerous applications in topology and, especially, in descriptive set theory (see, e.g., [891). L e m m a 4. Let X be a metric space, let Z be a subset of X and let
be a Borel mapping. Then there exist a set Z* C X and a mapping
satisfying the following relations: 1) z 2 z*; 2) Z* is a B o w l subset of X ; 3) f * is a B o w l mapping; 4) f * is an extension of f .
Proof. Taking account of the equality
we may try to apply here the method of transfinite induction. If our f belongs to the class Bao(Z, R ) , then we use the result contained in the previous lemma. Let now be a nonzero ordinal strictly less than wl, and suppose that the assertion is true for all functions belonging to the family u{BaC(Z,R) : C < 0.Let f be an arbitrary function from the
class Baf(Z, R). According to the definition of Bat (Z, R), there exist two sequences {& : . E N ) , {fn : n E N ) of ordinals and functions, respectively, such that a) for each n E N , we have
b) for any point
2;
L
E 2, we have
By the inductive assumption, for each n E N , there exist a Borel set X and a Borel function
extending
fn
(in particular, Z C 2:). Let us denote
2' = n{z; : n E N ) , Z* = { Z E Z' : there exists a limn++,
fi(2.)).
Evidently, the set Z* is Borel in X and Z G Z*. Further, for any point L E Z * , we may put f * ( ~=) limn-.+co f:(~). Then the mapping f * : Z* - R
defined in this manner is Borel, too, and extends the original mapping f . This finishes the proof of Lemma 4.
Remark 2. A slightly more precise formulation of Lemma 4 can be found, e.g., in monograph 1891 (the proof remains almost the same). But, for our further purposes, Lemma 4 is completely sufficient. Exercise 8. Let [O, 11'" denote, as usual, the Hilbert cube (or the Tychonoff cube of weight w ) . By applying the result of Lemma 4 , show that if X is a metric space, Z is a subset of X and a mapping
-
is Borel, then there exist a set Z* G X and a mapping g* :
z*
[O, 1IW,
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satisfying the following conditions: a) Z E Z*; b) Z* is a Borel subset of X; c) g* is a Borel mapping; d) g* is an extension of g. Now, we are ready to establish the result of Sierpiriski and Zygmund (in a more general form, not only for continuous restrictions of functions but also for Borel restrictions).
Theorem 2 . There exisis a function
such that, for any set Z E R of cardinality of the continuum, ihe restriction of f to Z is not a Borel mapping (from Z into R). In partacular, this resiriction i s not continuous. Proof. Let @ denote the family of all partial Borel mappings (from R into R ) defined on uncountable Borel subsets of R. Clearly, we have the equality card(@) = c where c denotes, as usual, the cardinality of the continuum. Consequently, we may enumerate this family in the following manner:
where a is the smallest ordinal with card(a) = c. Also, we can analogously enumerate the set of all points of R, i.e. represent R in the form R = {xp : ,8
< a).
Now, for each p < a, consider the family
of graphs of functions from {& : [
< P ) and take the set
Clearly, the latter set is not empty (moreover, it is of cardinality of the continuum). So we may pick a point (zp, yp) from this set. Let us put
Evidently, we obtain some mapping
Let us demonstrate that f is the desired function. Indeed, it immediately follows from the definition off that, for any partial Borel function q5 (acting from R into R) defined on an uncountable Borel subset of R, the inequality card({x E dom(4) : 4(z) = f (z)))
is fulfilled. Suppose that there exists a set Z E R of cardinality of the continuum, such that the restriction f lZ is Borel. Then, in conformity with Lemma 4, there exists a partial Borel mapping f * extending f (Zand defined on some uncountable Borel subset of R. Consequently, f * belongs to the family @, and
card({x E $om( f *) : f *(x) = f (z))) = c , which yields an obvious contradiction. Theorem 2 has thus been proved. Exercise 9. Give a generalization of Theorem 2 for uncountable Polish spaces. In other words, show that if E is an arbitrary uncountable Polish topological space, then there exists a function
such that, for each set Z C E of cardinality of the continuum, the restriction of f to Z is not a Borel mapping.
Remark 3. In a certain sense, we may say that any Sierpiriski-Zygmund function (defined on R) is totally discontinuous with respect to the family of all subsets of R having the cardinality of the continuum. In other words, such a function is very bad from the point of view of continuity of its restrictions to large subsets of R (here "large" means that the cardinality of those subsets must be equal to c). We shall see in the next chapter that a Sierpifiski-Zygmund function is also bad from the points of view of the Lebesgue measurability and the Baire property, i.e. such a function is not measurable in the Lebesgue sense and does not possess the Baire property. Actually, this fact follows directly from the Luzin theorem on the structure of Lebesgue measurable functions and from its corresponding analogue for the functions having the Baire property. Thus, we may conclude that the Sierpiriski-Zygmund construction yields an example of a function which
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simultaneously is not measurable in the Lebesgue sense and does not possess the Baire property. Of course, there are many other constructions of such functions (all of them are based on uncountable forms of the Axiom of Choice). The best known constructions are due to Vitali [I621and Bernstein [9]. We shall examine their constructions in our further considerations. R e m a r k 4. In connection with the Sierpiriski-Zygmund result presented above, the following question arises naturally: does there exist a function f : R+R such that, for any uncountable subset Z of R , the restriction f lZ is not continuous? As mentioned earlier, if the Continuum Hypothesis holds, then the Sierpiriski-Zygmund function yields a positive answer to this question. Nevertheless, the question cannot be resolved within the theory ZFC. Moreover, Shinoda demonstrated in [I301 that if Martin's Axiom and the negation of the Continuum Hypothesis hold, and g is an arbitrary function acting from R into R, then, for each uncountable set X C R, there always exists an uncountable set Y E X such that the restriction glY is continuous (for details, see [130]). Several related questions concerning the existence of a "good" restriction of a given function to some "nonsmall" subset of its domain are discussed in 1281 where the corresponding references can also be found.
6. Lebesgue nonmeasurable functions and functions without the Baire property
This chapter is devoted to some well-known constructions of a function nonmeasurable in the Lebesgue sense (respectively, of a function without the Raire property). Obviously, the existence of such a function is equivalent to the existence of a subset of the real line nonmeasurable in the Lebesgue sense (respectively, of a subset of this line without the Baire property). Since the fundamental concept of the Lebesgue measure on R (respectively, the concept of the Baire property) was introduced, it has been extremely useful in various problems of mathematical analysis. A natural question arose whether all subsets of R are measurable in the Lebesgue sense (and an analogous question was posed for the Baire property). Very soon, two essentially different constructions of extraordinary subsets of R were discovered which gave simultaneously positive answers to these questions. The first construction is due to Vitali [I621 and the second one to Bernstein [9]. Both of them were heavily based on an uncountable form of the Axiom of Choice, so it was reasonable to ask whether it is possible to construct a Lebesgue nonmeasurable subset of R (or a subset of R without the Baire property) by using some weak forms of the Axiom of Choice which are enough for most domains of the classical mathematical analysis (for instance, the Axiom of Dependent Choices). Almost all outstanding mathematicians working that time in analysis and particularly in the theory of real functions (Borel, Lebesgue, Hausdorff, Luzin, Sierpiriski, etc.) believed that there is no construction of a Lebesgue nonmeasurable subset of R within the theory ZF & DC. However, only after the corresponding developments in mathematical logic and axiomatic set theory and, especially, after the creation (in 1963) of the forcing method by Cohen, it became possible to establish the needed result. We shall return to this question in our further considerations and touch briefiy upon some related problems that are also interesting from the logical point of view. But first, we discuss more thoroughly analytic aspects of the problem of the existence of Lebesgue nonmeasurable sets (respectively, of sets without the Baire prop-
erty). Our starting point is one important common feature of Lebesgue measurable sets and of sets with the Baire property. We shall demonstrate that not all subsets of the real line have this feature. Obviously, this will give us the existence of required bad subsets of R. In order to carry out our program, we first introduce the following definition. Let X be a subset of R . We say that this set has the Steinhaus property if there exists a real E > 0 such that (Vh E R)((hl < E
* ( X + h ) n X # 0).
In other words, aset X C R has the Steinhaus property if the corresponding difference set X - X = {x' - x" : x' E X, x" E X ) is a neighbourhood of point 0. It turns out that, as a rule, all good subsets of R are either of Lebesgue measure zero, or of first category, or have the Steinhaus property. In this connection, it is reasonable to mention here that Steinhaus himself observed that all Lebesgue measurable sets on R with strictly positive measure have this property (see [154]). Some years later, it was also established that an analogous result is true for second category subsets of R having the Baire property. Let X denote, as usual, the standard Lebesgue measure on R. We now formulate and prove the following classical result.
Theorem 1. Let X be a subset of R satasfying at least one of these t w o assumpta'ons: 1) X E dom(X) and X(X) > 0; 2) X E Ba(R) \ K ( R ) . T h e n X possesses the Steinhaus property.
Proof. Suppose first that assumption 1) holds. Let x be a density point of X and let ]a, b[ be an open interval containing x for which we have
Obviously, there exists a strictly positive (Vh E R)((hl < E
E
such that
+ X(]a + h, b + h[ U ]a, b[) 5 4(b - a)/3).
Take an arbitrary h E R with J h (< E . We assert that
LEBESGUE NONMEASURABLE FUNCTIONS
Indeed, if ( X A(]a
+ h) n X = 0, then we must have
+ h, b + h[ U la,b[) 1 X(((Xn la, b [ ) + h) U ( x n ]a,b[))=
which is impossible. Thus X possesses the Steinhaus property. Suppose now that assumption 2) holds. Then X can be represented in the form X = unx, where U is a nonempty open set in R and X I is a first category subset of R.Evidently, there exists a real E > 0 such that (Vh E R ) ( ( h J< E
+ (U + h) n U # 0).
Let us fix any h E R with Ih( < E . It is easy t o check the inclusion
+
Taking account of the fact that (U h) fl U is a nonempty open subset of R and (XI h) U XIis a first category subset of R , we infer that
+
Consequently, (X+h)nx#0, and this finishes the proof of Theorem 1. The following statement is an easy consequence of Theorem 1 but, sometimes, is much more useful in practice.
Theorem 2. Let X and Y be subsets of R such that at least one of these two conditions holds: 1) {X, Y) c dom(X), X(X) > 0, X(Y) > 0; 2) { X , Y ) C B a ( R ) \ li'(R). Then the vector s u m
has a nonempty anteraor. Proof. Clearly, under assumption I), there exists an element t E R such that X((X t t ) n Y ) > 0.
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6
Actually, this relation follows from the metrical transitivity of the measure A (also, from the Lebesgue theorem on density points). Similarly, under assumption 2), there exists an element r E R such that
In fact, here we have the metrical transitivity for the Baire property. Let us put
z=(x+t)nr in the first case, and
Z=(X+r)nY in the second one. It suffices to show that the set Z + Z has a nonempty interior. If 2 is symmetric with respect to zero, then we may directly apply Theorem 1. Otherwise, we can find an element t E R such that
in the first case, and
in the second one. Finally, define
+
The set 2' is symmetric with respect to zero and Z' 2 Z z / 2 . Moreover, X(2') > 0 in the first case, and 2' E Ba(R) \ IC(R) in the second one. Applying Theorem 1 to 2' and taking account of the relation
we come to the required result, The following exercises show that Theorems 1 and 2 have analogues in much more general situations.
Exercise 1. Let E be an arbitrary topological space and let
be a family of first category open subsets of E. Prove, by applying the Zorn Lemma, that the set U=LJ{Ui : i E I )
LE BE S G U E N O N M E A S U R A BL E F UN C T I O N S
117
is of first category, too. This classical result is due to Banach (see, e.g., [89], [I101 or [117]). It is of some interest because the set of indices I may be uncountable here. Obviously, the assumption that all sets Ui are open in E is very essential for the validity of this result. Exercise 2. Let (GI .) be an arbitrary topological group and let X be a subset of G such that
Using the previous exercise, show that the set
is a neighbourhood of the neutral element of G. Deduce from this result that if A and B are any two subsets of G satisfying the relation
then the set A - B = { a e b : ~ E A ~, E B ) has a nonempty interior. Exercise 3. Let (G, .) be a a-compact locally compact topological group with the neutral element e and let p be the left invariant Haar measure on G. We denote by p1 the usual completion of p. Let X be an arbitrary p'-measurable subset of G. Starting with the fact that p is a Radon measure, prove that
In particular, if p'(X) > 0, then there exists a neighbourhood U ( e ) of e such that ('Jg E U ( ~ ) ) ( P / ( (-SX )n X ) > 0) and, consequently,
Conclude from this fact that if A and B are any two pl-measurable subsets of G with $(A) > 0 and pl(B) > 0, then the set A . B has a nonempty interior.
Now, we are ready to present the first classical construction of a subset of the real line, nonmeasurable in the Lebesgue sense and without the Baire property. As mentioned earlier, this construction is due to Vitali (see [162]). First, let us consider a binary relation Rv(x, y) on R defined by the formula X E R & ~ E R & X - ~ E Q where Q denotes, as usual, the set of all rational numbers. Since Q is a subgroup of the additive group of R, we infer that' Rv(x, y) is an equivalence relation on R.Consequently, we obtain the partition of R canonically associated with R v ( x , y). This partition will be called the Vitali partition of R and will be denoted by R / Q . Any selector of the Vitali partition will be called a Vitali subset of R. Theorem 3. There exist Vitali subsets o f R . I f X is an arbitrary Vitadi subset of R , then X is Lebesgue nonmeasurable and does not possess the Baire property. Proof. The existence of Vitali sets follows directly from the Axiom of Choice applied t o the Vitali partition. Let now X be a Vitali set and suppose for a moment that X is either Lebesgue measurable or possesses the Baire property. Then, taking account of the relation
we infer that X must be of strictly positive measure (respectively, of second category). But this immediately yields a contradiction. Indeed, for each rational number q # 0, we have
because X is a selector of R / Q . Here q may be arbitrarily small. In other words, we see that our X does not have the Steinhaus property. This contradicts Theorem 1. We thus obtained that Vitali sets are very bad from the points of view of the Lebesgue measure and the Baire property. However, these sets may be rather good for other nonzero a-finite invariant measures given on R . Exercise 4. Prove that there exists a measure p on the real line, satisfying the following conditions: a) p is a nonzero complete a-finite measure invariant under the group of all isometric transformations of the real line; b) dom(A) C dom(p) where A denotes the standard Lebesgue measure on R ;
LEBESOUE NONMEASURABLE FUNCTIONS
c) (VY E dom(A))(A(Y) = 0 + p(Y) = 0); d) (VY E dom(A))(A(Y)> 0 p(Y) = too); e) there is a Vitali set X such that X E dom(p). Moreover, since p is complete and a-finite, we can consider a von Neumann topology T(p) associated with p. Let R* denote the set of all real numbers, equipped with T(p). Then the a-ideal li'(R*) and the a-algebra Ba(R*) are invariant under the group of all translations of R* and the Vitali set X mentioned in e) belongs to Ba(R*), i.e. possesses the Baire property with respect t o T(p).
+
The next exercise shows that any Vitali set remains nonmeasurable with respect to each invariant extension of the Lebesgue measure. Exercise 5. Prove that, for any measure v on R invariant under the group Q and extending the Lebesgue measure A , no Vitali subset of R is v-measurable.
It is not hard to see that the argument used in the Vitali construction heavily relies on the assumption of the invariance of the Lebesgue measure with respect to translations of R . This argument does not work for those nonzero a-finite complete measures p on R which are only quasiinvariant (i.e. p is defined on a a-algebra of subsets of R , invariant under translations, and the a-ideal of all p-measure zero sets is preserved by translations, too). So the following question arises: how to prove the existence of nonmeasurable sets with respect to such a measure p . We shall consider this question in the next chapter of the book. Namely, we shall show there that a more general algebraic construction is possible yielding the existence of nonmeasurable sets with respect to p . The main role in that construction will be played by the so-called Hamel bases of R. Now, we want to turn our attention to another classical construction of a Lebesgue nonmeasurable set (of a set without the Baire property). As pointed out earlier, this construction is due to Bernstein (see [9]). First, let us introduce one useful notion closely related to the Bernstein construction. Let E be a topological space and let X be a subset of E . We say that X is totally imperfect in E if X contains no nonempty perfect subset of E. We say that X is a Bernstein subset of E if X and E \ X are totally imperfect in E. Equivalently, X is a Bernstein subset of E if, for each nonempty perfect set P C E, we have
It immediately follows from this definition that X C E is a Bernstein set if and only if E \ X is Bernstein. Clearly, each subset of the real line, having cardinality strictly less than the cardinality of the continuum, is totally imperfect. The question concerning the existence of totally imperfect subsets of the real line, having the cardinality of the continuum, turns out to be rather nontrivial. For its solution, we need uncountable forms of the Axiom of Choice (cf. the next exercise). Exercise 6. Prove, in the theory ZF & DC, that if there exists a totally imperfect subset of R of cardinality c, then there exists a Lebesgue nonmeasurable subset of R. Prove also an analogous fact for the Baire property. Exercise 7. Let n be a natural number greater than or equal to 2, and let X be a totally imperfect subset of the n-dimensional Euclidean space Rn.Show that the set R n \ X is connected (in the usual topological sense). Infer from this fact that any Bernstein subset of Rn is connected. There are many examples of totally imperfect subsets of the Euclidean space R n . A wide class of such sets was introduced and investigated by Marczewski (see [156]). Let E be a Polish topological space and let X be a subset of E. We say that X is a Marczewski subset of E if, for each nonempty perfect set P C_ E, there exists a nonempty perfect set P' C_ E such that
It immediately follows from this definition that every Marczewski set is totally imperfect in E, and that any subset of a Marczewski set is a Marczewski set, too. Also, it can easily be observed that any set Y E with card(Y) < c is a Marczewski set. Indeed, let us take an arbitrary nonempty perfect set P 5 E. Then, as we know (see Chapter I), there exists a disjoint family {Pa : i E I) consisting of nonempty perfect sets and satisfying the relations card(I)=c, (vi~I)(PicP).
c
Now, since card(Y)
< card(I), it is clear that there exists at least one index
io E I such that Pi, n Y = 0, and thus Y is a Marczewski set.
Let us recall the classical result of Alexandrov and Hausdorff stating that every uncountable Bore1 set in a Polish topological space contains a subset homeomorphic to the Cantor discontinuum (hence contains a nonempty perfect subset). Taking this result into account, we can give
12 1
LEBESGUE NONMEASURABLE FUNCTIONS
another equivalent definition of Marczewski sets. Namely, we may say that a set X lying in a Polish space E is a Marczewski set if, for each uncountable Borel subset B of E l there exists an uncountable Borel set B' C E satisfying the relations
B'EB,
B1nX=O.
In some situations, the second definition is more convenient. For instance, let El and E2 be two Polish spaces and let f : El
-+
E2
be a Borel isomorphism between them. Then, for a set X 5 El, the following two assertions are equivalent: 1) X is a Marczewski set in E l ; 2) f ( X ) is a Marczewski set in E2. In other words, the Borel isomorphism f yields a one-to-one correspondence between Marczewski sets in spaces El and E2. This fact is rather useful. For instance, suppose that we need to construct a Marczewski subset of El having some additional properties which are invariant under Borel isomorphisms. Sometimes, it turns out that such a set can much more easier be constructed in E2. Let us denote it by X'. Then we apply the Rorel isomorphism f-I to X' and obtain the required Marczewski set f q l ( X ' ) in the space E l . Later, we shall demonstrate the usefulness of this method. Namely, we shall show that there exist Marczewski subsets of R nonmeasurable in the Lebesgue sense (respectively, without the Baire property). One simple fact concerning Marczewski sets is presented in the next statement.
Lemma 1. Let { X k : k < w ) be a countable family of Marczewski subsets of a Polish space E. T h e n U { X k : k < w ) is a Marczewski set, too. I n particular, if the space E is uncountable, then the family of all Marczewski subseis of E forms a proper a-ideal i n the Boolean algebra of all subsets of E. Proof. Fix a nonempty perfect set P E E. Since Xo is a Marczewski set, there exists a nonempty perfect set Po E such that Poc P,
Pofl Xo = 0,
diam(Po) < 1.
Further, since X I is also a Marczewski set, there exist nonempty perfect sets Poo E E and Pol C E such that
CHAPTER 6
Proceeding in this manner, we will be able to define a dyadic system
{Pji...j b : jl = 0, j2
€ {O, I},
...,jk
€ {0, I}, 1 5 k
< W}
of nonempty perfect sets in E whose diameters converge t o zero, and
for each natural number k
> 1. Now, putting
we obtain a nonempty perfect set D E E satisfying the relation
This shows that U{Xk : k
< w} is a Marczewski subset
of E.
We thus see that, in an uncountable Polish topological space El the family of all Marczewski subsets of E forms a a-ideal. It is usually called the Marczewski u-ideal in E and plays an essential role in classical point set theory (cf. [17]). As mentioned above, Marczewski subsets of E can be regarded as a certain type of small sets in E . In our further considerations, we shall also deal with some other types of small sets which generate proper a-ideals in the basic set E. For instance, we shall deal with the a-ideal generated by all Luzin subsets of R (respectively, by all Sierpiliski subsets of R). In addition, we shall consider the a-ideals of the so-called universal measure zero subsets of R and of strongly measure zero subsets of R. Various properties of these subsets will be discussed in subsequent chapters of the book (note that valuable information about different kinds of small sets can be found in [17], [89] and [107]). Let us return to Bernstein sets. We now formulate and prove the classical Bernstein result on the,existence of such sets.
Theorem 4 . There exists a Bernstein subset of the real line. Ail such subsets are Lebesgue nonmeasurable and do not possess the Baare property. Proof. Let a denote the least ordinal number for which c a r d ( a ) = c. We know that the family of all nonempty perfect subsets of R is of
123
LEBESGUE NONMEASURABLE FUNCTIONS
cardinality c . So we may denote this family by {PC : E < a ) . Moreover, we may assume without loss of generality that each of the partial families
{PE : E < a, t is an even ordinal), {PC : E < a, ( is an odd ordinal) also consists of all nonempty perfect subsets of R. Now, applying the method of transfinite recursion, we define an a-sequence of points
satisfying the following two conditions: 1) if t < C < a, then xc # zc; 2) for each E < a , we have xe E PC. Suppose that, for P < a, the partial P-sequence already been defined. Take the set Pa.Obviously,
{XC : E < P ) has
Hence we can write
Pp \ {"E
:
E < PI # 0.
Choose an arbitrary element z from the last nonempty set and put x p = x. Continuing in this manner, we will be able to construct the a-sequence {xE : [ < a) of points of R, satisfying conditions 1) and 2). Further, we put X = {zF : ( < a, [ i s an even ordinal). It immediately follows from our construction that X is a Bernstein subset of R (because X and R \ X are totally imperfect in R). It remains t o demonstrate that X is not Lebesgue measurable and does not possess the Baire property. Suppose first that X is measurable in the Lebesgue sense. Then the set R \ X is Lebesgue measurable, too, and at least one of these two sets is of strictly positive measure. We may assume without loss of generality that X(X) > 0. Then a well-known property of X implies that there exists a closed set F C R contained in X and having a strictly positive .measure. Since X is a diffused (continuous) measure, we must have card(F) > w and hence card(F) = c. Denote by Fo the set of all condensation points of F . Obviously, Fo is a nonempty perfect subset of R contained in X . But this contradicts the fact that X is a Bernstein set in R.
124
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6
Suppose now that X possesses the Baire property. Then the set R \ X possesses the Raire property, too, and a t least one of these two sets is not of first category. We may assume without loss of generality that X is of second category. Consequently, we have a representation of X in the form
where V is a nonempty open subset of R and Y is a first category subset of R . Applying the classical Baire theorem, we,see that the set V \ Y contains an uncountable Gs-subset of R . This immediately implies that X contains also a nonempty perfect subset of R, which contradicts the fact that X is a Bernstein set in R . A result much more general than Theorem 4 is presented in the following exercise. Exercise 8. Let E be an infinite set and let {Xj : j E J) be a family of subsets of E, such that: 1) card(J) card(E); 2) (Vj E J)(card(Xj) = card(E)). Prove, by applying the method of transfinite recursion, that there exists a family {Yj : j E J) of subsets of E, satisfying the relations: a) ( V j E J)(Vjl E J ) ( j # j1 Yjn Yjl = 0); b) ( V j E J)(VjlE J)(card(Xj f l Y j l ) = card(E)).
<
+
Exercise 9. By starting with the result of the previous exercise, show that in every complete metric space E of cardinality of the continuum' (hence, in every uncountable Polish topological space) there exists a Bernstein set. Moreover, demonstrate that there exists a partition {Yj : j E J ) of E such that: a) card(J) = c; b) for each j E J, the set Yj is a Rernstein subset of E. Finally, show that if the space E has no isolated points, then all Bernstein subsets of E do not have the Baire property. The next exercise yields a characterization of Bernstein subsets of Polish spaces in terms of topological measure theory. Exercise 10. Let E be an uncountable Polish space and let X be a subset of E. Demonstrate that the following two assertions are equivalent: a) X is a Rernstein subset of E; b) for each nonzero a-finite diffused Bore1 measure p given on E, the set X is nonmeasurable with respect to the completion of p .
LEBESGUE NONMEASURABLE FUNCTIONS
125
Show also that these two assertions are not, in general, equivalent for a nonseparable complete metric space E. Exercise 11. Let us consider the first uncountable ordinal wl equipped with its order topology, and let
I = { X C wl
: ( 3 F C_ w l ) ( F i s closed, card(F)
= wl, F n X = 0)).
Show that I is a a-ideal of subsets of wl. The elements of I are usually called nonstationary subsets of wl. A set Z wl is called a stationary subset of wl if Z is not nonstationary. Let us put
where I' is the &filter dual to I. Observe that S is the a-algebra generated by I. Finally, demonstrate that, for any set X E wl, the following two relations are equivalent: a) the sets X and wl \ X are stationary in wl; b) for every nonzero a-finite diffused measure p defined on S, the set X is not measurable with respect to the completion of p. Any set X with the above-mentioned properties can be considered as an analogue (for the topological space wl) of a Bernstein subset of R. For more information about the a-ideal I and stationary subsets of wl, see [55], [56], [88] and [82]. The next exercise assumes that the reader is familiar with the notion of a complete Boolean algebra (for the definition, see e.g. [31]). Exercise 12. Let P ( R ) denote the complete Boolean algebra of all subsets of the real line R. Let Il be the a-ideal of all Lebesgue measure zero subsets of R and let I2be the a-ideal of all first category subsets of R. Consider the corresponding factor algebras P ( R ) / I l and P(R)/Iz. Show that these Boolean algebras are not complete. Exercise 13. Let E be an uncountable Polish space and let I be some a-ideal of subsets of E. We say that I has a Borel base if, for each set X E I, there exists a Borel subset Y of E such that XEY,
YEI.
Suppose that all one-element subsets of E belong to I and that I possesses a Borel base. Let us put S ( I ) = the a-algebra of subsets of E, generated by B ( E ) U I. Here B ( E ) denotes, as usual, the Borel a-algebra of the space E.
126
CHAPTER 6
Let X be a Bernstein subset of E (the existence of Bernstein sets in E follows, for instance, from Exercise 9). Show that X $ S ( I ) . Exercise 14. Prove that there exists a subset X of R which is simultaneously a Vitali set and a Bernstein set. All the constructions presented above were concerned with certain sets either nonmeasurable in the Lebesgue sense or without the Baire property. The existence of such sets evidently implies the existence of functions either nonmeasurable in the Lebesgue sense or without the Baire property. We now wish to consider a direct construction of a Lebesgue nonmeasurable function acting from R into R, An analogous construction can be carried out for the Baire property. In our further considerations, we denote by the symbol Az the standard two-dimensional Lebesgue measure on the Euclidean plane R ~Clearly, . Az is the completion of the product measure X x A where X denotes, as usual, the standard Lebesgue measure on the real line. We recall that a subset X of R 2 is X2-thick (or Az-massive) in R~ if, for each Az-measurable set Z 5 R2 with Xz(Z) > 0, we have
In other words, X is A2-thick in R~ if and only if the equality
is satisfied, where the symbol (A2), denotes the inner measure associated with X2. Let us point out that if a subset X of R 2 is Az-measurable and Azmassive simultaneously, then it is of full A2-measure, i.e.
Thus if we already know that a set X C R2 is not of full X2-measure but is Xz-thick, then we can immediately conclude that X is not X2-measurable. The next statement shows us that there are functions acting from R into R whose graphs are X2-thick subsets of the plane. T h e o r e m 5. There esists a jknction
whose graph is a Az-thick subset of R 2. Consequently, the followang two a.ssertions are true:
LEBESGUE NONMEASURABLE FUNCTIONS
1) the graph off as noi a A2-measur~blesubset of R ~ ; 2) f is not a A-me~surablefunction. Proof. Let a be the least ordinal number of cardinality continuum. Consider the family {Be : ( < a) consisting of all Bore1 subsets of R~ having strictly positive A2-measure. We are going to construct, by transfinite recursion, a family of points
satisfying these two conditions: (1) if < C < a, then xe # X C ; (2) for each I < a, the point (xe, ye) belongs to Be. Suppose that, for an ordinal P < a, the partial family
has already been defined. Let us take the set Bp. Applying the classical Fubini theorem, we see that the set
is A-measurable and has a strictly positive measure. Consequently, this set is of cardinality of the continuum, and there exists a point x belonging to it and distinct from all the points xe (( < P ) . We put xp = x. Then we choose an arbitrary point y from the set Bp(xp) and put yp = y. Proceeding in this manner, we will be able to construct the required family {(xe, ye) : ( < a). Now, it easily follows from condition (1) that the set
can be regarded as the graph of a partial function acting from R into R. We extend arbitrarily this partial function to a function acting from R into R and denote the latter function by f . Then condition (2) implies that the graph o f f is Az-thick in R 2 . Since there are continuum many pairwise disjoint translates of this graph, we conclude that the graph is not of full Az-measure and hence it is not a Az-measurable subset of R 2 . Finally, the function f is not A-measurable. Indeed, otherwise the graph of f will be a Az-measure zero subset of the plane, which is impossible. This ends the proof of the theorem. Exercise 15. By applying the Kuratowski-Ulam theorem which is a topological analogue of the classical Fubini theorem (see, e.g., [89] or [117]),
prove a statement for the Baire property, analogous to Theorem 5. Namely, show that there exists a function
such that its graph is thick in the sense of the Baire property, i.e, the graph intersects each subset of R 2 having the Baire property and not belonging to the a-ideal of all first category subsets of R2, Deduce from this fact that the graph of f does not have the Baire property in R a and f does not have the Baire property as a function acting from R into R. Exercise 16. Theorem 5 with the previous exercise show us that there exist functions from R into R whose graphs are thick subsets of the plane (in particular, those graphs are nonmeasurable in the Lebesgue sense or do not have the Baire property). On the other hand, prove that there exists a measure p on R 2 satisfying the following conditions: a) p is an extension of the Lebesgue measure X 2 ; b) p is invariant under the group of all translations of R2 and under the central symmetry of R~ with respect to (0,O); c) the graph of any function acting from R into R belongs to dom(p) and, for any such graph I?, we have p ( r ) = 0. We thus see that the graphs of all functions acting from R into R are small with respect to the above-mentioned measure p. This is a common property of all functions from R into R. Another interesting common feature of all functions from R into R was described by Blumberg's theorem (see Chapter 5 of this book). If we deal with some class of subsets of R which are small in a certain sense, then, as a rule, it is not easy to establish the existence of a set belonging to this class and nonmeasurable in the Lebesgue sense (or without the Baire property). Theorem 5 yields us that there exist functions from R into R whose graphs are nonmeasurable with respect to A 2 . At the same time, as mentioned above, the graphs of such functions may be regarded as small subsets of R 2 with respect to the measure p of Exercise 16. More generally, suppose that a a-ideal I of subsets of R is given. Then the following natural question can be posed: does there exist a set X E I nonmeasurable in the Lebesgue sense or without the Baire property? Obviously, the answer to this question depends on the structure of I and simple examples show that the answer can be negative. Let us consider the two classical a-ideals: I(X) = the a-ideal of all A-measure zero subsets of R;
L EBE SG U E N O N ME A S U R A BL E F U N C TI O N S
129
K ( R ) = the a-ideal of all first category subsets of R. These two a-ideals are orthogonal, i.e. there exists a partition { A , B ) of R such that A E I(A), B E II'(R). The reader can easily check this simple fact which immediately implies the existence of a Lebesgue nonmeasurable set belonging to K ( R ) and the existence of a Lebesgue measure zero set without the Baire property. Indeed, let X be a Bernstein subset of R. We put
Then it is easy to check that: 1) Xo E I(A) and Xo does not possess the Baire property; 2) X I E K ( R ) and XIis not measurable in the Lebesgue sense. A more general result is presented in the next exercise.
Exercise 17. Let X be a Bernstein subset of R. Let Y be a Ameasurable set with A(Y) > 0 and let Z be a subset of R having the Baire property but not belonging to I<(R).Show that the set X n Y is not measurable in the Lebesgue sense and that the set X f l Z does not possess the Baire property. Returning to the question formulated above, we wish to consider it more thoroughly for the Marczewski a-ideal in R. In other words, it is natural to ask whether there exist Marczewski subsets of R nonmeasurable in the Lebesgue sense (or without the Baire property). This problem was originally raised by Marczewski (see [156]). The solution to it was independently obtained by Corazza [32]and Walsh [164]. Here we want to present their result. First of all, we need one auxiliary proposition useful in many situations. The proof of this proposition is very similar to the argument used in the proof of Theorem 5.
Lemma 2. Let {Zj : j E J ) be a family of subsets of the plane R 2 , such that: 1) card(J) c; 2) for each index j E J, the set of all x E prl(Zj) satisfying the relation
<
as of cardinality of the continuz6m. Then there exist a set-valued mapping
and an injective family { x j : j E J)C_R, such that, for any index j E J, we have the equalities
Proof. Obviously, we may assume without loss of generality that card(J) = c. Also, we can identify the set J with the least ordinal number a for which card(&)= c . Now, we are going to define a set-valued mapping F and a family { x F : ( < a ) by the method of transfinite recursion. Suppose that, for an ordinal /3 < a , the partial families
have already been constructed. Consider the set Zp. According to our assumption, the set of all those x E p r l ( Z p )for which
is of cardinality continuum. Since
there exists a point x E R such that
Therefore, we can put
In this way, we will be able to define F and { x e : ( < a ) with the required properties. By utilizing the previous lemma, it is not difficult to establish the following statement.
Theorem 6. There exists a Marczewski subset of R2 nonmeasurable in the Lebesgue sense and without the Baire property. Proof. Let a be again the least ordinal number with card(&)= c and let { Z t : E < a ) denote the family of all Bore1 subsets of R 2 having a strictly positive A2-measure. Applying Lemma 2 to this family, we can find a set-valued mapping F and an injective family { x t : ( < a ) of points of
13 1
LEBESGUE NONMEASURABLE FUNCTIONS
<
R with the corresponding properties. Let now {PC: < a) be the family of all nonempty perfect subsets of R 2 . For each P < a, we put
Note that yp can be defined because of the equality card(F(P)) = c. Let us check that
is a Marczewski set nonmeasurable with respect to X2. Indeed, Do can be regarded as the graph of a partial function acting from R into R.From the construction of Do we have that Do is also a A2-thick subset of R2. Consequently (cf. the proof of Theorem 5), we may assert that Do is nonmeasurable in the Lebesgue sense. It remains to show that Do is a Marczewski set. In order to do it, take an arbitrary nonempty perfect subset P of Ra. We must verify that P contains a nonempty perfect set whose intersection with Do is empty. If there exists at least one point x E R for which card(P(x)) = c, then there is nothing to prove. Suppose now that
For some p < a, we have Pp = P. Then, taking account of the definition of Ye (E < 4, we get
The latter relation easily implies that there exists a nonempty perfect subset of Pp which does not have common points with Do. In a similar way, starting with the family of all those Bore1 subsets of the plane R 2 which do not belong to the a-ideal Ii'(R2), we can construct a Marczewski set Dl C R2 thick in the category sense. Then it is not hard to see that
D=DoUD1 is a Marczewski set in R 2 nonmeasurable in the Lebesgue sense and without the Baire property. Exercise 18. Show that there exists a Marczewski subset D of R 2 such that: 1 ) D does not possess the Baire property;
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2) D is not measurable with respect to the completion of any product measure p x v where p and v are Borel probability measures on R vanishing on all one-element subsets of R. We can easily infer from Theorem 6 the existence of Marczewski subsets of R nonmeasurable in the Lebesgue sense (respectively, without the Baire property). Indeed, consider a Borel isomorphism
It is a well-known fact that 4 can be chosen in such a way that, simultaneously, 4 will be an isomorphism between the measures A and A2. Now, , 4-l(X) is a if X is a A2-nonmeasurable Marczewski subset of R ~then A-nonmeasurable Marczewski subset of R. Analogously, a Borel isomorphism 4 can be chosen in such a way that it will preserve the Baire category of sets (consequently, the Baire property of sets). Now, if X is a Marczewski subset of R~ without the Baire property, then 4 - l ( X ) is a Marczewski subset of R without the Baire property. A stronger result is contained in the next exercise. Exercise 19. Prove that there exists a Borel isomorphism
such that: a) $J preserves the category of sets (in particular, $J preserves the Baire property); b) is an isomorphism between the measures A and A2. By starting with this fact and applying Theorem 6, show that there exists a Marczewski subset of the real line, nonmeasurable in the Lebesgue sense and without the Baire property. $J
Our last example of a Lebesgue nonmeasurable function acting from R into R, which also does not have the Baire property, can be obtained by using the corresponding results of Chapter 5. Let us recall the theorem of Sierpiriski and Zygmund proved in Chapter 5 and stating that there exists a function
satisfying the following condition: for each subset X of R with card(X) = c, the restriction f lX is not continuous on X . By starting with this condition, it is not hard to show that f is not Lebesgue measurable and does not possess the Baire property.
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Indeed, suppose first that f has the Baire property. Then, according to a well-known theorem (see, e.g., [89]), we can find a dense Gs-subset A of R such that f ) A is continuous. Obviously,
and we obtain a contradiction with the fact that f is a Sierpiliski-Zygmund function, Now, let us demonstrate that f is not measurable in the Lebesgue sense. We will prove a much more general result asserting that f is not measurable with respect to the completion of any nonzero a-finite diffused Borel measure on R. Let p be such a measure and let p' denote the completion of p. Since an analogue of the classical Luzin theorem holds true for p', we can find a closed subset B of R with pl(B) > 0 such that the restricted function f lB is continuous. Taking account of the diffusedness of p' and of the inequality p1(B) > 0, we infer that B is uncountable and hence card(B) = c. This yields again a contradiction with the fact that f is a Sierpiriski-Zygmund function.
Exercise 20. Let Ii' be a compact subset of R2.Obviously, the set p r l ( I < ) is compact in R.Show that there exists a Borel mapping
such that the graph of 4 is contained in Ii'. In addition, give an example of a compact subset P of R 2 for which there exists no continuous mapping
such that the graph of $ is contained in P. This simple result is a very particular case of much more general statements about the existence of measurable selectors for set-valued mappings measurable in various senses. For example, suppose that a Borel subset B of the Euclidean plane R~is given satisfying the relation
Can one assert that there exists a Borel function
such that its graph is contained in B? Luzin and Novikov (see, e.g., [97]) showed that, in general, the answer to this question is negative, i.e, there
are Borel subsets B of R 2 with prl(B) = R which do not admit a Borel uniformization la. On the other hand, suppose that we have an analytic subset A of R 2 and consider its first projection prl(A) which is an analytic subset of R.Then, according to the classical theorem of Luzin, Jankov and von Neumann (see, for instance, [65]), there exists a function
such that: 1) the graph of g is contained in A; 2) g is measurable with respect to the a-algebra generated by the family of all analytic subsets of R. In particular, one may assert that the above-mentioned function g has the Baire property in the restricted sense and is measurable with respect to the completion of any a-finite Borel measure given on R. This important theorem has numerous applications in modern analysis and probability theory (some such applications are presented in [65]). Exercise 21. By using the result of the previous exercise, show that the graph of any Sierpiriski-Zygmund function is a Marczewski subset of the Euclidean plane. Exercise 22. Construct, by using the method of transfinite recursion, a Sierpiriski-Zygmund function whose graph is a &-thick subset of the Euclidean plane. Applying a similar method, construct a Sierpiriski-Zygmund function whose graph is a thick subset of the plane in the category sense. Deduce from these results that there are Marczewski subsets of the plane, nonmeasurable in the Lebesgue sense (respectively, without the Baire property). On the other hand, show that there exists a Sierpiriski-Zygmund function whose graph is a A2-measure zero subset of the plane. Analogously, show that there exists a Sierpiriski-Zygmund function whose graph is a first category subset of the plane. Concluding this chapter, we wish to make some remarks about logical aspects of the question concerning the existence of a Lebesgue nonmeasurable subset of the real line (or of a subset of the same line without the Baire property). Namely, in 1970, Solovay published his famous paper [I511 where he pointed out a model of ZF & DC in which all subsets of the real line were Lebesgue measurable and possessed the Baire property. However, the existence of such a model was based on the assumption of the existence of an uncountable strongly inaccessible cardinal number and this seemed to be
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a weak side of the above-mentioned result. But, later, Shelah showed in his
remarkable work [I291 that large cardinals appeared here not accidentally. More precisely, he established that: 1) there are models of ZF & D C in which all subsets of R possess the Baire property; 2) the existence of a model of ZF & D C in which all subsets of R are Lebesgue measurable implies the existence of some large cardinal. Solovay constructed also another model of set theory in which all projective subsets of R are Lebesgue measurable and possess the Baire property (see [151]). In this connection, it is reasonable to recall that in the Constructible Universe of Gijdel there are projective subsets of R (even belonging to the class P r 3 ( R ) ) which are not Lebesgue measurable and do not have the Baire property (for more details, see e.g. [55] and 1561). From among many other results connected with the existence of sets nonmeasurable in the Lebesgue sense (respectively, of sets without the Baire property), we want to point out the following ones: 1. Kolmogorov showed in [83] that the existence in the theory ZF & D C of a universal operation of integration for all Lebesgue measurable functions acting from [ O , l ] into R implies the existence of a Lebesgue nonmeasurable function acting from R into R. A similar result is true (in the same theory.) for a universal operation of differentiation. We thus conclude that the two fundamental operations of mathematical analysis - integration and differentiation - lead directly to real-valued functions which are nonmeasurable in the Lebesgue sense. This result seems to be interesting and important from the point of view of foundations of real analysis. In Chapter 13, we shall present some statements concerning generalized derivatives, which are closely related to the above-mentioned Kolmogorov result. 2. Sierpiriski proved that the existence of a nontrivial ultrafilter in the Boolean algebra of all subsets of w implies (within the theory ZF & D C ) the existence of a subset of R nonmeasurable in the Lebesgue sense and without the Baire property.
The proof of this result can be found, e.g., in [26]. 3. Shelah and Raisonnier (cf. 11201) established that the implication wl
holds in
=+ there exists a Lebesgue nonmeasurable set on R
ZF & D C .
4. Pawlikowski showed in [I181 that the Hahn-Banach theorem on extensions of linear functionals implies (in ZF & DC) the famous Banach-Tarski paradox and, hence, implies the existence of a Lebesgue nonmeasurable subset of R. Various topics related to the Banach-Tarski paradox are discussed in monograph 11631 by Wagon. In particular, certain parts of this monograph are specially devoted to deep relationships between equidecomposability theory and the problem of the existence of Lebesgue nonmeasurable subsets of R. Some other results concerning the existence of nonmeasurable sets and of sets without the Baire property are presented in [24], [25], [28], [40], [55], [561, [571, [681, [691, P O I [801, 1861, [1021, [ l l O I J [11711-[1341, [i3-61J-11381, [144], 11491, [I501 and [163].
7. Hamel basis and Cauchy functional equation
In this chapter we discuss some properties of Hamel bases of the real line R and show their remarkable role in various constructions of strange additive functions acting from R into R. The existence of such a basis was first established by Hamel in 1905 (see [54]). Later, it was also shown that the existence of a Hamel basis cannot be proved without the aid of uncountable forms of the Axiom of Choice (in particular, it is impossible to establish the existence of such a basis within the theory ZF & DC). The construction of a Hamel basis can be done by starting with one general theorem of the theory of vector spaces (over arbitrary fields). Let us recall that, according to this general theorem, for every vector space E, there exists a basis of E, i.e, a maximal (with respect to inclusion) linearly independent subset of E. This assertion follows almost immediately from the Zorn Lemma, i.e, from the Axiom of Choice. So, each element of E can be represented as a linear combination of some elements of the basis and such a representation is unique. Let us now consider the real line as a vector space over the field Q of all rational numbers. Then, applying the above-mentioned general theorem, we get that there are bases of R over Q . Any such basis is called a Hamel basis of R.
Exercise 1. Let E be an arbitrary vector space. Show that any two bases of E have the same cardinality. Their common cardinality is called the algebraic (or linear) dimension of the space E. Notice that the result presented in Exercise 1 is a very particular case of a general theorem about the cardinality of any system of free generators of a free universal algebra (in this connection, see e.g. 1311). Let {es : i E I) be an arbitrary Hamel basis of R. It is clear that
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As we know, every element x of R may be uniquely represented in the form
~ ~ ~indexed family of rational numbers, such that where ( q i ( ~ ) is) some
For each index i E I, the rational number qi(x) is called the coordinate of x corresponding to this index. From the purely group-theoretical point of view, this representation simply means that the additive group of the real line can be regarded as the direct sum of a family of groups {Qi : i E I), satisfying the following conditions: 1 ) card(1) = c; 2 ) for each index i E I, the group Qi is isomorphic to the additive group of Q . In other words, we may write
Clearly, an analogous representation will be true for the n-dimensional Euclidean space R n , where n 1, for an infinite-dimensional separable Hilbert space and, more generally, for an arbitrary vector space over Q having the cardinality of the continuum. Thus, from the group-theoretical point of view, all these spaces are isomorphic (i.e. all of them are isomorphic to the group CiElQi). This observation seems to be very trivial. However, by adding some argument, it yields rather nontrivial consequences. We wish to consider here one of such consequences. Let us recall that the existence of Lebesgue nonmeasurable subsets of the real line was shown in the previous chapter of the book. More precisely, we discussed there the two classical constructions of nonmeasurable sets, due to Vitali and Bernstein, respectively. The first of them was based on certain algebraic properties of the Lebesgue measure X (namely, on the invariance of this measure under the group of all translations of the real line) and the second one was based on some topological properties of A (actually, we essentially used the fact that this measure is Radon). Als'o, it was mentioned in the previous chapter that the Vitali construction cannot be generalized to the class of all nonzero a-finite measures
>
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given on the real line and quasiinvariant with respect to the group of all translations of this line (we recall that a nonzero mesure p given. on R is quasiinvariant under the group of all translations of R if don+) and the a-ideal generated by the family of all p-measure zero sets are invariant under this group). In other words, the argument used in the Vitali construction does not work for nonzero a-finite quasiinvariant measures. In this connection, the following two related problems arise: Let ,u be a nonzero a-finite measure on R quasiinvariant with respect to the group of all translations of R. 1. Does there exist a subset of R nonmeasurable with respect to p? 2. Does there exist a subgroup of R nonmeasurable with respect to p? Our purpose here is to establish that the answer to the second question is positive (consequently, the answer to the first question turns out to be positive, too). In the process of establishing this fact we shall essentially use the existence of a Hamel basis of the real line (hence an uncountable form of the Axiom of Choice). In our further considerations, we need several auxiliary statements. These statements are not difficult to prove. We shall see that they lead us to the required result.
Lemma 1. Let G be a commutative group and let Go be a szbbgroup of G. Let p be a a-finite G-quasiinvariant measure on G. Suppose also that: 1) the factor group GIGo is uncountable; 2) p*(Go) > 0 where p-enotes, as usual, the outer measure associated with p. Then Go is nonmeasurable with respect to ,u. Proof. Suppose for a moment that Go E dom(p). Then, according to condition 2), we must have the inequality
On the other hand, according to condition I), there is an uncountable family of pairwise disjoint G-translates of Go in G. Taking account of the a-finiteness and G-quasiinvariance of our measure p , we infer that the equality GO) = 0 must be true, which contradicts the above-written inequality. The contradiction obtained ends the proof of Lemma 1. The following exercise contains a slightly more general result than the preceding lemma.
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Exercise 2. Let G be an arbitrary group and let X be a subset of G. Suppose that G is equipped with a a-finite left (respectively, right) G-quasiinvariant measure p and suppose also that these two relations are fulfilled: 1) there exists an uncountable family of pairwise disjoint left (respectively, right) G-translates of X in G; 2) p* (X) > 0. Show that the set X is nonmeasurable with respect to p . Lemma 2. Let G be again a commutative group and let {G, : n E w ) be a countable family of subgroups of G. Suppose that the following two conditions are satisfied: 1) for each natural number n, the factor group GIG, is uncountable; 2) U{G, : n E w ) = G. Further, lei p be a nonzero a-finite G-quasiinvariant measure on G. Then there exists at least one n E w such that the corresponding group G, is nonmeasurable with respect to p. Proof. Since p is not identically equal to zero and the given family {G, : n E w ) is a countable covering of G, there exists a natural number n such that pr(Gn) > 0. Now, applying Lemma 1 to the subgroup G,, we see that this subgroup is nonmeasurable with respect to p . Lemma 3. Let G be a commutative group, let H be a proper subgroup of G and let I be an anfinite set of indices. Consader two direct sums of Gi and CiEIHi where, for each index i E I , the group Gi groups coincides with G and the group Hi coincides with H . Then we have
In partacular, if the set I is uncountable, then the factor group
is uncountable, too.
Proof. Since H is a proper subgroup of G, there exists an element D the family of all those elements
g E G \ H. Let us denote by the symbol
HAMEL BASIS A N D CAUCHY FUNCTIONAL EQUATION
which satisfy the relation
Obviously, we have
c a r d ( D ) = card(1). It is not hard to check that if dl and dl1 are any two distinct elements from D ,then
Consequently, we obtain the required inequality
and the proof of Lemma 3 is completed.
Lemma 4. The additive group of R can be represented i n the form
where {G, : n E w ) is a countable family of subgroups of R and, for each n E w , the factor group RIG, is uncozbntable. Proof. First, let us consider the additive group Q and show that this group can be represented as the union of a countable increasing (with respect to inclusion) family of its proper subgroups. Indeed, for any natural number n , define
k Q'"'={~ : kEZ}. Evidently, Q(,) is a subgroup of Q and, since there exist prime natural numbers strictly greater than n!, this subgroup is proper. Moreover, we may write (Vn)(Vna)(n 5 na < w =+ Q(,) E Q ( ~ ) ) , i.e. the family of groups {Q(,) inclusion. Also, it is clear that
: n
< w) is increasing with respect to
As shown earlier (with the aid of a Hamel basis), we have the representation of R in the form of a direct sum
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where I is a set of indices with
and, for each i E I, the group Qi coincides with Q . Now, for any natural number n, we put
It can easily be shown (by applying Lemma 3) that the family {G, : n E w ) of subgroups of R is the required one. The preceding lemmas immediately give us the following generalization of the classical Vitali theorem. T h e o r e m 1. For any nonzero a-finite measure p on R quasianva~iant under the group of all translations of R, there esasts a subgroup of R nonmeasurable wath respect t o p. The next exercise contains a more general result. Exercise 3. Let E be an uncountable vector space over the field Q . Prove that there is a countable family {I?, : n E w ) of subgroups of the additive group of E , satisfying the following condition: for any nonzero afinite measure p on E quasiinvariant under the group of all translations of E, there exists at least one n E w for which the group ,'I is nonmeasurable with respect to p . R e m a r k 1. By applying some methods of infinitary combinatorics (namely, by using the famous transfinite matrix of Ulam [161]), it can be shown that, for any nonzero a-finite measure p on R quasiinvariant under the group of all translations of R, there exists at least one vector subspace of R (over the field Q , of course) nonmeasurable with respect to p . Moreover, a similar result can be established for uncountable commutative groups. Namely, for every uncountable commutative group G and for any nonzero a-finite G-quasiinvariant measure v on G, there exists a subgroup of G nonmeasurable with respect to v. Notice that, in order to obtain such a result, we need not only the methods of infinitary combinatorics mentioned above, but also ,some deep theorems concerning the algebraic structure of infinite commutative groups (for more details, see [73]). Notice also that similar statements hold true (for uncountable commutative groups again) in terms of the Baire property (see [73]). On the other hand, there are examples of uncountable noncommutative groups for which analogous results fail to be true.
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Let us return to Hamel bases of the real line. These bases were found not accidentally but as a tool for a solution of a concrete question in mathematical analysis. Now, we are going to formulate this question and discuss its close relationships with Hamel bases of R. Let us consider the class of all those functions
which satisfy the following functional equation:
This equation is usually called the Cauchy functional equation. Notice that this equation simply says that f is a homomorphism from the additive group of R into itself. Also, it is easy to see that any homomorphism f from the additive group of R into itself satisfies the relation
for all x E R and for all q E Q . In other words, f can be regarded as a linear mapping when R is treated as a vector space over the field Q . The problem is to find all solutions of the Cauchy functional equation. It is clear that there are very natural solutions of this equation. Namely, every function f : R + R of the form
where a is a fixed real number, is a solution of this equation. Such solutions we shall call trivial ones (let us stress that any continuous solution of the Cauchy functional equation is trivial). Harnel bases allow us to construct nontrivial solutions of this equation. Namely, we have the following result due to Hamel (see [54]).
Theorem 2. There are nontrivial solutions of the Cauchy functional equataon. Proof. Let {ed : i E I) be an arbitrary Hamel basis of R. As was noticed earlier, any x E R can be uniquely represented in the form
Let us fix an index ia E I and define a function
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by the following formula:
It is clear that 4 satisfies the Cauchy functional equation. Moreover, 'the range of this function is contained in Q . Also, this function is not constant since 4(0) = 0, 4(ei,,) = 1, So 4 is not a continuous function (we shall see below that 4 is not even Lebesgue measurable and does not possess the Baire property). Theorem 2 has thus been proved.
Exercise 4. Show that there exists a nontrivial automorphism of the additive group of R onto itself. How many such automorphisms are there? The next statement yields a characterization of nontrivial solutions of the Cauchy functional equation in terms of their graphs.
Theorem 3. Let f be a solution of the Cazbchy functional equation. Then the following two assertions are equivalent: 1) the graph o f f is dense everywhere in the plane R ~ ; 2) f is a nontrivial solution of the Cauchy functional equation. Proof. Obviously, if f is a trivial solution of the Cauchy functional equation, then the graph of f , being a straight line, is nowhere dense in quently, if a solution of the Cauchy functional equation has the R ~Conse . graph dense in R 2 , then this solution must be nontrivial. We thus see that implication 1) =+ 2) is valid. Let us establish the converse implication 2) =+ 1). In order to do it, suppose that 2) is satisfied but the graph of f is not dense everywhere in R 2. Then there exists a nonempty open rectangle
such that
Gf
n (la, b[xlc,d[) = 0
where Gf denotes, as usual, the graph of f . Let us show that at least one of the following two relations holds: (1) for all x E ]a, b [ , we have f(x) 2 c; (2) for all x E ]a, b[, we have f ( x ) _< d.
HAMEL BASIS A N D CAUCHY FUNCTIONAL EQUATION
Indeed, assuming that there are two points ]a, b[, satisfying the inequalities
XI
and xz from the interval
we can easily infer that, for some rational numbers ql ql qz = 1, the relation
+
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> 0 and qz > 0 with
will be fulfilled, i.e.
which is impossible. The contradiction obtained yields that a t least one of relations (1) and (2) must be true. We may suppose, without loss of generality, that relation (2) holds. Now, it is easy to see that there exist a strictly positive number 6 and a real number h, such that
Taking relation (2) into account and applying the additivity of our function f , we deduce that f is bounded from above on the open interval ] - 6,6[. Since (a: E R ) , f(-.) = -f(x) we also infer that f is bounded on the same interval and, consequently, is continuous at point 0. Finally, using the additivity o f f again, we conclude that f is continuous a t all points of R, which contradicts assumption 2). This contradiction establishes the converse implication 2) =+1). Theorem 3 has thus been proved.
Exercise 5. Let E be an arbitrary normed vector space over the field of real numbers (or over the field of complex numbers) and let E have an infinite algebraic dimension. Show that there exists a linear functional defined on E and discontinuous a t each point of E. Compare this result with the fact that every linear functional defined on a finite-dimensional normed vector space is continuous. Suppose now that a function f : R -+ R is given satisfying the Cauchy functional equation. Then, as mentioned above, for every x E R and for every q E Q , we have f ( q . 2 ) = P.f(.).
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It immediately follows from this fact that if f is a continuous function a t least a t one point of R, then
for each x E R. This result was first established by Cauchy. It is also easy to prove a slightly more general result stating that if f is a solution of the Cauchy functional equation and, in addition, has an upper (or lower) bound on some nonempty open interval, then f is a trivial solution of this equation (cf. the proof of Theorem 3). Hence we see that nontrivial solutions of the Cauchy functional equation are very bad from the topological point of view; namely, they are discontinuous a t each element of their domain. The next result (due t o Frechet) shows us that nontrivial solutions of the Cauchy functional equation are also bad from the points of view of the Lebesgue measure and the Baire property.
Theorem 4. No nontrivial soleiion of the Caechy functaonal eqeation is Lebesgee measurable (respectively, possesses the Baire property). Proof. There are many proofs of this remarkable theorem. We shall present here a very simple one based on the Steinhaus property for the Lebesgue measure (respectively, for the Baire category). Suppose that a function f : R-R is given satisfying the Cauchy functional equation and suppose that f is Lebesgue measurable (respectively, has the Baire property). Let us consider the sets of the form f -I([-n, nl> (n E 4. All these sets are Lebesgue measurable (respectively, have the Baire property) and u,,,f-'([-n, n]) = R. Hence there exists a natural number n such that
or, respectively, f-I([-., n]) E B a ( R ) \ Ii'(R). Then the Steinhaus property for the Lebesgue measure (respectively, its analogue for the Baire category) implies that the set
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is a neighbourhood of the neutral element of R . Let now V be any open interval in R such that
Then we get
f ( V ) c [-2n, 2nl1
so the function f is bounded on V. From this fact it immediately follows that f is continuous at point 0. Hence f is a trivial solution of the Cauchy functional equation, and the proof of Theorem 4 is completed. As mentioned earlier, there are also many other proofs of the preceding theorem which are based on essentially different ideas and methods. One purely analytic proof of this theorem was suggested by Orlicz (see, e.g., [114]). However, the argument presented above (used the Steinhaus property for measure and category) seems to be more natural and can be applied in situations where a given Lebesgue measurable function (or function hav-
ing the Baire property) is not necessarily additive (see, for instance, the proof of Theorem 5 below). Considering Theorems 2 and 4, we can conclude that the existence of a Hamel basis in R implies the existence of a Lebesgue nonmeasurable subset of R and the existence of a subset of R without the Baire property. More exactly, we see that the proof of these two implications can be done in the theory ZF & DC.
Exercise 6. Let {es : i E I ) be a Hamel basis of R . Fix an index io E I and put r = the vector space (over Q) generated by {ei : i E I \ { i o ) ) .
In other words, r is a hyperplane in R regarded as a vector space over Q. Show in the theory ZF & DC that: a) the real line can be covered by a countable family of translates of I?; b) r does not have the Steinhaus property, i.e, for each E > 0, there exists a rational number q E 10,E[ such that
Deduce from relations a) and b) that I? is not measurable in the Lebesgue sense and does not possess the Baire property. Conclude from this result that the implication
(there exists a Hamel basis of R ) =+ (there exists a subset of R nonmeasurable in the Lebesgue sense
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and w.ithout the Baire property) is a theorem of the above-mentioned theory. E x e r c i s e 7. By using the theorem on the existence of a Hamel basis of R, describe all solutions of the Cauchy functional equation. E x e r c i s e 8. Describe all continuous functions
which satisfy the following functional equation:
Show that there are discontinuous (and Lebesgue nonmeasurable) solutions of this functional equation, too. Describe all such solutions. One can prove that there are Lebesgue measurable and Lebesgue nonmeasurable Hamel bases (see exercises below). This fact shows us an essential difference between Hamel bases and Vitali sets from the point of view of Lebesgue measurability. The same is true for the Baire property. E x e r c i s e 9. Prove that there are two sets A C R and B C R both of Lebesgue measure zero and of first category, such that
In particular, we have the equality (AUB)
+
(AUB)=R
where the set A U B is also of Lebesgue measure zero and of first category. Starting with this fact, show (by applying the method of transfinite induction) that there exists a Hamel basis in R which is contained in the set A U B and, consequently, is of first category and of Lebesgue measure zero. In addition, show that, for every a-finite measure p on the real line which is invariant (or, more generally, quasiinvariant) under the group of all translations of R and for every Hamel basis { e i : i E I), we have the implication {ei :
i E I) E dom(p) =+p({ei
:
i E I}) = 0.
E x e r c i s e 10. By using the method of transfinite induction, prove that there exists a Hamel basis in R which simultaneously is a Bernstein subset
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of R . Conclude that such a Harnel basis is A-nonmeasurable (where A denotes the standard Lebesgue measure on R ) and does not possess the Baire property. R e m a r k 2. In connection with the results presented in the two preceding exercises, the following question arises naturally: is any Hamel basis of R totally imperfect in R ? It turns out that the answer to this question is negative. Namely, it can be shown that there are nonempty perfect subsets of R linearly independent over the field Q (see, for instance, [I101 or [I631 where a much stronger statement is discussed with its numerous applications). Let P be such a subset. Then, by using the Zorn Lemma, one can easily prove that there exists a Harnel basis of R containing P. Obviously, this Hamel basis will not be totally imperfect. The following question also seems to be natural: if p is a nonzero o-finite measure on the real line R , invariant under the group of all translations of R , does there exist a Hamel basis in R nonmeasurable with respect to p? As pointed out above (see Exercise lo), the answer is positive for the classical Lebesgue measure on R. However, in the general case, this question is undecidable in the theory ZFC. The next exercise presents a more precise result concerning the question. Exercise 11. Prove that the following two statements are equivalent: a) the Continuum Hypothesis ( c = w l ) ; b) for every nonzero a-finite measure p given on the real line R and invariant under all translations of R, there exists a Hamel basis nonmeasurable with respect to p . In connection with Exercise 11, see [39] and [134]. In our considerations above, we were primarily concerned with the Cauchy functional equation
Let us now consider the weakest form of the Jensen inequality which plays a basic role in the theory of real-valued convex functions. We recall that a function
satisfies the Jensen inequality (or is midpoint convex) if
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for all x E R and y E R. A more general form of the Jensen inequality which is used in classical mathematical analysis is the following one:
for all x E R and y E R and for all positive real numbers ql, qz whose sum is equal to 1. It is well known that this more general form of the Jensen inequality describes the class of all convex functions acting from R into R. This class is very important from the point of view of applications. Also, functions belonging to this class have rather nice properties. For instance, they are continuous and possess a derivative almost everywhere. Moreover, a classical theorem from analysis states that any convex function possesses a derivative at all points of R except countably many of them, and possesses also a second derivative at almost all points of R (indeed, the derivative of a convex function is monotone, so the desired result follows at once from the Lebesgue theorem on differentiability almost everywhere of any monotone function). At the same time, the weakest form of the Jensen inequality, written above, does not restrict essentially the class of admissible functions. Indeed, i f f is a solution of the Cauchy functional equation, then f obviously satisfies the weakest form of the Jensen inequality. But we know that f may be a nontrivial solution of the Cauchy functional equation and, in this case, f is even Lebesgue nonmeasurable (and does not have the Baire property). In this connection, the following result due to Sierpiriski (and generalizing Theorem 4) is of some interest.
Theorem 5. Suppose that
is a Lebesgue measurable function (respectively, a funclion with the Baire property) satisfying the inequality
Then f is continuous and, consequently, f is a convex function in the usual sense.
Proof. The argument is very similar to the one used in the proof of Theorem 4. First of all, we may assume without loss of generality that f ( 0 ) = 0. Otherwise, it is sufficient to introduce a new function f l by the formula f l ( 4 = f(.) - f (0) ( 8 E R).
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151
Evidently, this new function is also Lebesgue measurable and satisfies the weakest form of the Jensen inequality. So, in our further considerations, we suppose that f (0) = 0. In particular, for each x E R, we have
and, consequently, -f(-8) I f (XI. Now, for any natural number n , we consider the set
Xn = f-'(] - O O , ~ ] ) . According t o our assumption, all these sets are measurable in the Lebesgue sense (respectively, possess the Baire property). Since
R = U{X, : n E w } , a t least one of these sets is not of Lebesgue measure zero (respectively, is not of first category). Let X, be such a set. Then the Steinhaus property implies that the vector sum
has a nonempty interior and hence contains some nondegenerate open interval ]a, b [ . Now, the inequality f(x)if(y) 2
(x E R , y E R )
applied to Z, immediately yields that
i.e, our function f is bounded from above on the interval ]a/2, b/2[. In addition, the inequality
shows us that f is bounded from above on any interval of the form
Consequently, f is bounded from above on each subinterval of R with the compact closure and, actually, f turns out t o be locally bounded on R.
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Let now t be an arbitrary point of R and let h be a nonzero real number such that lhl < 1. Obviously, we may write
From this inequality, applying an easy induction on k, we get
But it is clear that, for some natural number k, we have
Let us denote Lo = Lo(t1 f ) =
SUP,e[t-l,t+l]
f (XI.
Then the preceding inequalities imply f (t
where L1 relation
+ h) - f ( t ) I (Lo - f (t)) -21h1 = L1 . IhI
= Ll(t, f ) 1 0. Using an analogous argument, we come to the f(t)-f(t+h)
I Lz .lhl
with L2 = Lz(t,f ) 2 0. Finally, we obtain
where L = L(t, f) 2 0 is some constant. In this way we have proved that our function f is continuous a t the point t and hence f is continuous a t all points of the real line (since t was taken arbitrarily). In other words, f is a convex function in the usual sense. An extensive account of the Cauchy functional equation and the weakest form of the Jensen inequality is presented in the monograph by Kuczma P61. There are also many textbooks and special monographs devoted to the theory of convex sets and convex functions (this theory is usually called convex analysis). Here we wish to present a few additional facts about convex functions, interesting from the purely analytical point of view. We shall do it in the next two exercises.
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Exercise 12. Let f and g be two functions acting from R into R. Suppose that, for all points x E R and y E R and for each number p E [0, 11, the inequality
is satisfied. Consider the set
Let conv(I'*(g))denote the convex hull of I'*(g). Show that c o n v ( r * ( g ) ) coincides with the union of the family of all those triangles whose vertices belong t o the set I'*(g). Show also that the boundary of conv(I'*(g))can be regarded as the graph of some convex function
for which the relation
is fulfilled. In other words, the convex function 4 separates the given functions f and g .
Exercise 13. Let f be a function acting from R into R and let E be a strictly positive real number. We say that f is an &-convex function if, for any points x E R and y E R and for each number q E [O,l], we have
By applying the result of the previous exercise, demonstrate that i f f is an &-convex function, then there exists a convex function r$
such that
: R-tR
f ( 4 < d ( x ) 5 f(.)
+
( x E R).
Furthermore, define a function
11 by the formula
( 1 =4
: R-tR
-1
(2 E
R)
CHAPTER 7
and show that (x E R ) . I$(.> - f(x)I 5 €12 In other words, for every &-convexfunction f , there exists a convex function t,6 such that the norm [I$ - f 11 is less than or equal to &/2. This result is due to Hyers and Ulam (see [59]; some related results and problems are also discussed in [160]). Let now f be a function acting from R into R and suppose that, for some E > 0, we have the relation
In this case, we cannot assert, in general, that there exists a convex function t,6 acting from R into R and satisfying the inequalities
for each x E R.Indeed, i f f is a nontrivial solution of the Cauchy functional equation, then f is a Lebesgue nonmeasurable function and, obviously, it cannot be uniformly approximated by convex functions (which are measurable in the Lebesgue sense). The next exercise contains some information about the descriptive structure of a Hamel basis of the real line.
Exercise 14. Let X be an arbitrary Hamel basis in R. By starting with the fact that any analytic subset of R is Lebesgue measurable and applying the Steinhaus property for Lebesgue measurable sets, prove that X is not an analytic subset of R. This old classical result is due to Sierpiriski. In connection with it, let us remark that there are uncountable linearly independent (over Q) subsets of R having a good descriptive structure. For instance, as mentioned earlier, there are nonempty perfect subsets of R linearly independent over Q (see [I101 and [163]). Let f be a function acting from R into R and let x be a point of R. We say that f is symmetric with respect to x if
for all t E R. In other words, f is symmetric with respect to x if its graph is a symmetric subset of R 2 with respect to the straight line {x) x R.
HAMEL BASIS A N D CAUCHY FUNCTIONAL EQUATION
155
We say that a function f : R -c R is almost symmetric with respect to x € R if card({t E R : f ( x - t ) # f (x t ) ) ) < c.
+
It is easy to see that if f is symmetric with respect to each x E R, then f is constant. At the same time, by using a Hamel basis of the real line, it can be proved that there exist almost symmetric functions with respect to all x E R, which are not constant. This result is due to Sierpiriski (see [141]). More precisely, Sierpiriski established that the following statement is true.
Theorem 6 . There exists a Lebesgue nonmeasurable (and without the Baire property) function f almost symmetric with respect to all points of R. In particular, such a function is not equivalent to a constant function, i.e. there exists no set Z C R of Lebesgue measure zero (of first category) such that the restrictaon o f f to R \ Z is constant. Proof. Let a denote the least ordinal number for which card(a) = c. Let {PC : E < a } be the family of all nonempty perfect subsets of R. We may assume without loss of generality that each of the two partial families {PC : {PC :
E < a , $, i s a n even ordinal}, E < a , E is a n odd ordinal)
consists of all nonempty perfect subsets of R, too. Now, by applying the method of transfinite recursion, it is not hard to define a family
{el :
4
of elements of R, such that: a) el E Pt for each 4 < a ; b) {el : 4 < a ) is linearly independent over Q . This construction is very similar to the classical Bernstein construction, so we leave the corresponding details to the reader. Further, we can extend the family {el : E < a } to some Hamel basis of R. This Hamel basis will be denoted by the symbol { e g : ( < a ) . Obviously, we may suppose that each of the sets
Eo = {eC : 4 El = {et : 4
< a , 4 i s an even ordinal}, < a, 4 i s an odd ordinal)
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7
is a Bernstein subset of R (hence these two sets are nonmeasurable in the Lebesgue sense and do not possess the Baire property). Now, for any real number r, we have a unique representation of r in the form
where n = n ( r ) is a natural number, all q l , pa, ..., qn are nonzero rational numbers and
ti < E2 < < En. We define a mapping
by the formula
4 ( r ) = etn
(T
ER\
{O)).
Further, we put
X = { r E R : $ ( r ) E Eo}. Finally, let us denote by f the characteristic function of X . Clearly, f is Lebesgue nonmeasurable and does not possess the Baire property. Fix a point x E R and let t be an arbitrary point of R \ ( 0 ) . If x # 0, then we may write $(XI = et, $ ( t ) = ec
<
for some ordinal numbers E < cu and < a. If 4 < C, then we have the relation X - t ~ X w z + t ~ X . This relation is also true for x = 0 . Now, taking account of the equality
<<
E}) denotes the vector space (over Q) where the symbol lin({ec : generated by {eC : 5 0,we immediately obtain that
<
card({t E R : f ( x
- t) # f ( x + t ) } )< c,
i.e, our function f turns out to be almost symmetric with respect to x. Since x was taken arbitrarily, we conclude that f is almost symmetric with respect to all points of R. Theorem 6 has thus been proved. Some additional information around Theorem 6 can be found in [45].
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Exercise 15. Let f be a function acting from R into R and let x E R . Let A denote the standard Lebesgue measure on R. We say that f is Aalmost symmetric with respect to x i f f is X-measurable and
Show that if f is A-almost symmetric with respect to all points belonging to some everywhere dense subset of R, then f is A-equivalent to a constant function. Formulate and prove an analogous result in terms of the Baire property. Hamel bases have many other interesting and important applications and not only in analysis. One of the most beautiful applications may be found in the geometry of Euclidean space, more precisely, in the theory of polyhedra lying in this space. We mean here that part of this theory which is connected with the Hilbert third problem about the equivalence (by a finite decomposition) of a three-dimensional unit cube and a regular threedimensional simplex with the same volume. The Hilbert third problem will be briefly discussed in the last exercise of this chapter. It is reasonable to mention, in this connection, that Hamel was one of Hilbert's many students and worked in the geometry of Euclidean space, as well.
Exercise 16. Let us recall that a (convex) polyhedron in the space R~ is an arbitrary convex subset of this space which can be represented as the union of a nonempty finite family of closed three-dimensional simplices. For any two polyhedra X C R 3 and Y C R 3 , we say that they are equivalent by a finite decomposition if there exist two finite families {Xi : i E I},
{Yi
: i€I}
of polyhedra, such that: a) X = U i E I Xi, Y = U j E I b) for all distinct indices i E I and if E I, we have
where the symbol int denotes, as usual, the interior of a set; c) for each index i E I, the polyhedra Xi and Yi are congruent with respect to the group of all motions (i.e. isometric transformations) of the space R 3 . Let f : R -+ R be any solution of the Cauchy functional equation, such that f(?r) = 0. Let P3 denote the class of all polyhedra in R 3 and let us define a functional :
P3 -+R
CHAPTER
7
by the formula
C
f ( a j ) ,1411 ( X E Pa), jeJ where (bj)jEJ is the injective family of all edges of the polyhedron X and aj is the value of the dihedral angle of X corresponding to the edge bj (i.e. aj is the value of the angle which is formed by two faces of X meeting at the edge bj) and, finally, Ibj 1 denotes the length of the edge b j s Show that the functional cPf is invariant under the group of all motions of the space R3 and has equal values for any two polyhedra equivalent by a finite decomposition. We say that cPf is a Dehn functional on P3 associated with the solution f of the Cauchy functional equation. Let a be the value of the dihedral angle corresponding to an edge of the regular three-dimensional simplex. Show, by applying the Zorn Lemma, that the set { a , n} can be extended to a Hamel basis in R. Conclude from this fact that there exists a solution (XI =
of the Cauchy functional equation,such that g(a) = 1 and g(n) = 0. Show that the functional cPg assigns a strictly positive value on the three-dimensional regular simplex of volume 1 and, simultaneously, cPg assigns the value zero on the closed unit cube in R 3 . Hence these two polyhedra, being of the same volume, are not equivalent by a finite decomposition. This gives us the solution of the Hilbert third problem, first obtained by his disciple Dehn (see [33]). Remark 3. The result presented in the last exercise explains why, in the course of elementary geometry, we need to use some infinite procedures or limit processes for calculation of the volume of a three-dimensional simplex. After the Hilbert third problem was solved, another problem arose naturally concerning necessary and sufficient conditions for equivalence by a finite decomposition of two given polyhedra. This essentially more difficult problem was finally solved by Sydler who proved that two given polyhedra X and Y in the space R3 are equivalent if and only if they have the same volume and, for every Dehn functional as,the equality @,(X> = @/(Y) is fulfilled (see the original paper by Sydler [155]; cf. also [62] where a much simpler argument is presented). This topic and a number of related questions are thoroughly considered in the excellent textbook by Boltjanski 1121.
8. Luzin sets, Sierpihski sets and their applications
In this chapter, our discussion is devoted to the so-called Luzin subsets of the real line R and the Sierpiliski subsets of the same line. These sets are useful in various questions of real analysis and measure theory. Also, they have a number of applications in modern set theory (in particular, in constructing some special models of ZFC). First of all, we wish to emphasize the fact that the existence of Luzin and Sierpiriski subsets of the real line cannot be established within the theory Z F C , so if we want to deal with such subsets, then we need some additional set-theoretical axioms (see, for instance, Theorems 1 and 4 below). We begin our considerations with some properties of Luzin sets. These sets were constructed by Luzin, in 1914, under the assumption of the Continuum Hypothesis. The same sets were constructed by Mahlo one year before Luzin. However, in the mathematical literature the notion of a Luzin set is usually used, probably because Luzin investigated these sets rather deeply and showed a number of their important applications to the theory of real functions and classical measure theory (see, for example, ~981). We now give the precise definition of Luzin sets. Let X be a subset of the real line R . We say that X is a Luzin set if 1) X is uncountable; 2) for every first category subset Y of R , the intersection X fl Y is a t most countable. It is obvious that the family of all Luzin subsets of the real line R generates the a-ideal invariant under the group of all those transformations of R which preserve the a-ideal of all first category subsets of R (consequently, the above-mentioned family is invariant with respect to the group of all homeomorphisms of the real line and, in particular, with respect to the group of all translations of this line). We remark at once that it is impossible to prove in the theory Z F C the
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existence of a Luzin set. Namely, assume that the relation
(Martin's Axiom) & (2, > wl) holds and take an arbitrary set X 5 R. If X is at most countable, then it is not a Luzin set. Suppose now that X is uncountable. As known (see, e.g., [88]), Martin's Axiom implies that the union of an arbitrary family : i E I) of first category (respectively, of Lebesgue measure zero) subsets of R , where card(I) < c , is again of first category (respectively, of Lebesgue measure zero). In particular, under Martin's Axiom, each subset of R with cardinality strictly less than c is of first category and of Lebesgue measure zero. Let now Y be any subset of X of cardinality w l . Then, taking account of the inequality w l < c , we infer that Y is a first category subset of R and
{x
so we obtain again that X is not a Luzin set. Fortunately, if we assume the Continuum Hypothesis, then we can demonstrate that Luzin sets exist on the real line R (we recall that this classical result is due to Luzin and Mahlo).
Theorem 1. If the Continuum Hypothesas holds, then there are Luzin subsets of the real line R. Proof. As we know, the Continuum Hypothesis means the equality c = w l , which implies, in particular, that the family of all Borel subsets of R has cardinality w l . Let (Xt)tc,, denote the family of all first category Borel subsets of R. We define, by the method of transfinite recursion, a family (xg)gcwl of points from R. Suppose that E < w l and that the partial family (xC)CcShas already been defined. Let us consider the set
It is clear that Zt is a first category subset of the real line. Hence, by the classical Baire theorem, there exists a point
Thus we may put xc = x. This ends the construction of the family ( ~ g ) ~ <., , Now, we define X = {xt : ( < w l ) .
Notice that if C
< < < w l , then x c # xc. Therefore, we get
Suppose now that Z is an arbitrary first category subset of the real line. Then there exists an ordinal ( < w l such that Z E Xt (because the family (Xt)t<,, forms a base of the a-ideal of all first category subsets of R). Obviously, we have
and we see that X is a Luzin set. This finishes the proof of Theorem 1. Notice that if X is a Luzin set on R, then the set X U Q is also a Luzin set. Hence, the Continuum Hypothesis implies that there are Luzin sets everywhere dense in R.We can easily get a much stronger result. Namely, a slight modification of the proof presented above gives us a Luzin set X such that, for any set Y E R with the Baire property, we have the relation
(card(Y fl X )
< w) @ (Y i s o f
first category).
Exercise 1. By assuming the Continuum Hypothesis and applying the method of transfinite induction, show that there exists a Luzin subset X of R such that, for each set Y E R possessing the Baire property, the following two relations are equivalent: a) card(Y fl X ) w ; b) Y is a first category subset of R.
<
R e m a r k 1. It is reasonable to mention here that the existence of Luzin subsets of R is also possible in some cases when the Continuum Hypothesis does not hold. Namely, there are certain Cohen-type models of set theory in which the negation of the Continuum Hypothesis is valid and there exist Luzin sets of cardinality c (for details, see e.g. [87] and [88]). Notice, in addition, that in those models we also have a subset of R of cardinality wl < c , which does not possess the Baire property (cf, the next exercise). Exercise 2. Let X be a Luzin set on R with c a r d ( X ) = c and let n be a cardinal number satisfying the relations
Show that no subset Y of X with c a r d ( Y ) = K. possesses the Baire property in R. Luzin sets have a number of specific features important from the point of view of applications in analysis. The following theorem was also proved by Luzin.
Theorem 2. Suppose that X is a Luzin set on R. Then X does not possess the Baire property in R (and, moreover, any uncountable subset of X , being also a Luzin set, does not possess the Baire properiy). Furthermore, an the space X U Q equipped with the topology induced by the standard topology o f R every first category set is at most countable and, conversely, every at most countable subset of the space X U Q as of first category an
XUQ. Proof. Let X be an arbitrary Luzin set on R . Suppose, for a moment, that X has the Baire property. Since X is uncountable and, for every first category set Y C R , we have
we infer that X is not a first category subset of R . But then X contains some uncountable G6-subset 2. Let Y be any subset of Z homeomorphic to the classical Cantor discontinuum. Then Y is nowhere dense in R and Y E X ,We also have
which contradicts the definition of a Luzin set. The contradiction obtained yields us that X does not have the Baire property. The second part of this theorem follows from the fact that X U Q is a Luzin set, too, and, in addition, X U Q is everywhere dense in R . The proof of Theorem 2 is thus completed. The result proved above shows us that if the Continuum Hypothesis holds, then there exists an uncountable topological space X C R everywhere dense in R and such that
where Ii'(X) denotes, as usual, the a-ideal of all first category subsets of the space X and [XISWdenotes the family of all (at most) countable subsets of X. The latter equality also implies that
where B a ( X ) denotes the class of all subsets of X with the Baire property and B ( X ) denotes the Borel a-algebra of X . By taking this result into account, it is reasonable to introduce the following definition. A Hausdorff topological space E without isolated points is called a Luzin space if the equality Ii'(E) = [ E ] < ~ is valid. Hence we see that, under the Continuum Hypothesis, there exists a dense Luzin set on the real line R, which can be regarded as an example of a topological Luzin space. In our further considerations, we need one simple auxiliary statement concerning first category supports of a-finite diffused Borel measures given on separable metric spaces. Recall that a measure p defined on a u-algebra of subsets of a set E is diffused (continuous) if, for each e E El we have { e ) E dom(p) and p ( { e ) ) = 0.
Lemma 1. Let E be an arbitrary separable metrac space and let p be a u-finite dzffused Borel measure on E . Then there exisis a subset Z of E such that: I ) Z E I<(E) and Z is an F,-subset of E ; 2) p ( E \ Z) = 0. In other words, Z is a first category suppori o f p . Proof. Without loss of generality, we may assume that p is a probability measure, i.e. p(E) = 1. Let us denote by {en : n < w} a countable dense subset of the given space E . Fix a natural number k . Since our p is a diffused measure, we can find, for each point e n , an open neighbourhood V k ( e n )such that p ( v k ( e n ) ) < 1/2ktn. Let us put Vk = U { V , ( e n ) : n < w ) . Then Vk is a dense open subset of E l for which we have Now, putting
Y = n{vk
: k
< w),
we obtain a dense Gs-subset Y of E of p-measure zero. Consequently, Z = E \ Y is a first category F,-subset of E such that
Thus Z is the required support of our measure p .
Exercise 3. Let E be a topological space satisfying the following conditions: 1) there exists a countable subset D of E everywhere dense in E; 2) each one-element set { e ) , where e E D , is a Gpsubset of E. Let p be an arbitrary a-finite Borel measure on E such that
Prove that there exists a first category F,-subset Z of E for which
Obviously, this result is a slight generalization of Lemma 1. The next theorem, also essentially due to Luzin, shows us an interesting connection between Luzin sets on R and topological measure theory.
Theorem 3. Let X be an arbitrary Luzin subset of R. Then the following two relations hold: 1) af p is any u-finate diffused Borel measure on R, then p*(X) = 0 where p* denotes the outer measure associated with p; 2) zf p is any a-finite diffused Borel measure on ihe topological space X , then p is identically equal to zero. Proof. It is almost obvious that relations 1) and 2) are equivalent. Hence it is sufficient to prove only the second one. Without loss of generality, we may assume that X is a dense subset of R. Let p be an arbitrary u-finite diffused Borel measure given on the topological space X. Since X is a separable metric space, we may apply the preceding lemma on first category supports of diffused Borel measures. According to this lemma, there exists a first category subset of X on which our p is concentrated. But we know that each first category subset of X is at most countable, and our p is diffused, so we conclude that p must be identically equal to zero. Theorems 2 and 3 show us that, on the one hand, from the topological point of view Luzin sets are extremely pathological (because any uncountable subset of a Luzin set does not have the Baire property) but, on the other hand, from the point of view of topological measure theory Luzin sets are very small, since they have measure zero with respect to the completion of any u-finite diffused Borel measure on R.
Exercise 4. Let E be a topological space such that all one-element sets { e ) , where e E El are Borel (in E).We shall say that E is a universal
measure zero space (or universally negligible space) if each a-finite diffused Borel measure given on E is identically equal to zero. It immediately follows from Theorem 3 that, under the Continuum Hypothesis, there exist universal measure zero subspaces of the real line, having the cardinality of the continuum (namely, any Luzin subset of R is such a space). In particular, the Continuum Hypothesis implies that c is not a real-valued measurable cardinal (for the definition of real-valued measurable cardinals and their properties, see [55], [56], [82], [88], [117], [I321 and ~1521). Show that: a) each subspace of a universal measure zero space is also a universal measure zero space; b) the class of all universal measure zero spaces is closed under finite Cartesian products, but is not closed under countable Cartesian products; c) if I is a set of indices, whose cardinality is not real-valued measurable, and {Ed : i E I ) is a family of universal measure zero spaces, then the topological sum of {Ed : i E I } is a universal measure zero space, too.
Exercise 5. Let E be a topological space satisfying the assumption of the previous exercise. Prove that the following two assertions are equivalent: a) E is a universal measure zero space; b) for any topological space E' satisfying the same assumption and containing E as a subspace, and for any a-finite diffused Borel measure p on El, we have the equality
where p* denotes, as usual, the outer measure associated with p . Deduce from this result that if {En : n E w ) is a countable family of universal measure zero subspaces of a topological space El, then the space U{En : n E w } is universal measure zero, too. In other words, the family of a11 universal measure zero subspaces of El forms a a-ideal in the Boolean algebra of all subsets of E' (this a-ideal is proper if El is not itself universal measure zero).
Exercise 6. Let El and Ez be two topological spaces satisfying the assumption of Exercise 4, and let
be an injective Borel mapping. Check that if the space Ez is universal measure zero, then the space El is universal measure zero, too.
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Show also that the condition of injectivity o f f is essential here. Moreover, show that a continuous image of a universal measure zero space is not, in general, universal measure zero.
Exercise 7. Prove that any universal measure zero subspace X of the real line R is a Marczewski subset of R , i.e, for each nonempty perfect set P R , there exists a nonempty perfect set P' C R such that PIC
P,
plnx=O.
As mentioned above, any Luzin subset of the real line is universal measure zero. But the existence of Luzin sets cannot be proved in the theory ZFC. On the other hand, it is known that the existence of uncountable universal measure zero subsets of R can be established within ZFC (see, for instance, [89] or [107]). One of the earliest examples of an uncountable universal measure zero subset of R was constructed by Luzin who applied, in his construction, some specific methods of the theory of analytic sets. In fact, the construction of Luzin yields a universal measure zero set Y C R with card(Y) = w l . In particular, we may conclude that it is impossible to establish within the theory ZFC that Y is a Luzin set in the sense of the definition presented in this chapter. In addition, let us remark that Luzin's result concerning the cardinality of universal measure zero subsets of R is rather precise. Namely, as shown by Baumgartner and Laver (cf. [107]), in the Solovay-type model of set theory each universal measure zero subspace of R has cardinality less than or equal to w l . As we see, Luzin sets and universal measure zero sets on R may be regarded as small subsets of R . There are also many other notions of a small subset of the real line. One of such notions was introduced by Borel in 1919. Let X be a subset of R . We say that X is small in the Borel sense (or is a strongly measure zero set) if, for any sequence {E, : n < w ) of strictly positive real numbers, there exists a countable covering { ]an,bn[ : n < w ) of X by open intervals, such that (Vn
< w)(bn - a, < E,).
It immediately follows from this definition that the family of all subsets of R small in the Borel sense forms a u-ideal in the Boolean algebra of all subsets of R .
Exercise 8. Show that any Luzin subset of the real line is small in the Borel sense.
Exercise 9. Let [a, b] be an arbitrary compact subinterval of R and let p be a finite diffused Borel measure on [a, b]. Show that, for each real E > 0, there exists a real 6 > 0 such that, for any subinterval It', t"] of [a, b] with t'' - t' < 6, we have p([t',iU]) < E . Deduce from this fact that any subset of the real line, small in the Borel sense, is universal measure zero (note that the converse assertion is not true in the theory ZFC). Assuming the Continuum Hypothesis and taking account of the result of Exercise 8, we have the existence of uncountable subsets of R small in the Borel sense. The Borel Conjecture is the following set-theoretical statement: every set on R small in the Borel sense is at most countable. Thus, under the Continuum Hypothesis, the Borel Conjecture is false. However, Laver demonstrated that there are some models of set theory in which this conjecture holds true (for more detailed information, see [I071 and the corresponding references therein). Let us return to Luzin subsets of the real line. The next two exercises contain some additional information about these sets. Exercise 10. Let X be a Luzin subset of the real line R and let p be a a-finite diffused Borel measure on R.Suppose also that
is a mapping which has the Baire property. Prove that
where p* denotes, as usual, the outer measure associated with p. Hence, f ( X ) is a universal measure zero subset of R. Exercise 11. Suppose that the Continuum Hypothesis holds and let X be an uncountable dense subspace of the real line R, such that
Show that X is a Luzin subset of R . A dual (in a certain sense) object to a Luzin set is the so-called Sierpiriski set which was constructed by Sierpiliski, also under the assumption of the Continuum Hypothesis, in 1924.
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Let us introduce the notion of a Sierpiriski set and consider some properties of such sets. Let X be a subset of R . We say that X is a Sierpiliski set if 1) X is uncountable; 2) for each Lebesgue measure zero subset Y of R , the intersection X n Y is at most countable. Many facts concerning Sierpiriski sets are very similar to the corresponding facts concerning Luzin sets. For example: a) the family of all Sierpiriski sets generates the a-ideal of subsets of the real line R, invariant under the group of all those transformations of R which preserve the a-ideal of all Lebesgue measure zero subsets of R (in particular, this family is invariant with respect to the group of all translations of R ) ; b) the assumption
(Martin's Axiom) & (2W> w l ) implies that there are no Sierpiliski sets on the real line. Analogously, we have the following theorem due to Sierpiliski.
Theorem 4. Assume the Continuum Hypothesis. Then there exist Sierpin'ski subsets of the real line R . The proof of this theorem is very similar to the proof of Theorem 1. The only change is the replacement of the family (Xe)E<w,of all first category Borel subsets of R by the family (YE)(<,, of all A-measure zero Borel subsets of R (where X denotes, as usual, the standard Lebesgue measure on R ) . Also, instead of the classical Baire theorem, we must apply here the trivial fact that X is not identically equal to zero.
Exercise 12. Give a detailed proof of Theorem 4. In connection with Theorem 4, we wish to note that the existence of Sierpiriski subsets of R is possible in some situations when the Continuum Hypothesis does not hold. More precisely, there are models of ZFC in which the negation of the Continuum Hypothesis is valid and there exist Sierpiliski sets of cardinality c (see, e.g., [87] and [88]). Evidently, in such models we also have Lebesgue nonmeasurable subsets of R with cardinality Wl < c.
Remark 2. There is a general result due to Sierpiliski and Erdos which states that, under certain additional set-theoretical hypotheses (in particular, under the Continuum Hypothesis or Martin's Axiom), the a-ideal of all
first category subsets of R is isomorphic to the a-ideal of all Lebesgue measure zero subsets of R.An isomorphism between these two classical a-ideals is purely set-theoretical and does not have good properties. However, the existence of such an isomorphism enables us to obtain automatically many theorems for the Lebesgue measure (the Baire category) starting with the corresponding theorems for the Baire category (the Lebesgue measure). In particular, in this way we can easily deduce Theorem 4 from Theorem 1 (and, conversely, Theorem 1 from Theorem 4). A detailed proof of the Sierpiriski-Erdos result mentioned above (the socalled Sierpiliski-Erdos Duality Principle) is given in the well-known textbooks [I101 and [I171 where numerous applications of this principle are presented as well (a general version of the Duality Principle is formulated and proved in [26]).
Exercise 13. Let T denote the density topology on R . Show that a set Z E R is a Sierpiliski set in R if and only if Z is a Luzin set in the space ( R , T ) (this means that Z is uncountable and, for every first category set Y in ( R , T ) , the intersection Z fl Y is at most countable). Let us point out another similarity between Luzin and Sierpiliski sets.
Theorem 5. Every Sierpan'ski set is a first category subset of the real lane R . No uncountable subset of a Sierpin'skz set is Lebesgue measumb!e. Proof. Let X be a Sierpiliski set. Let I(X) denote, as usual, the a-ideal of all Lebesgue measure zero subsets of R . As we know, the a-ideals IC(R) and I(X) are orthogonal, i.e, there exists a partition {A, B ) of R such that
Simultaneously, we have the inequality
and the inclusion XcAU(XnB), from which we immediately obtain that X E Ii'(R). Suppose now that Y is an uncountable subset of a Sierpiliski set X (hence Y is a Sierpiliski set, too). Since X n Y is uncountable, we observe that Y $2 I(X). Suppose, to the contrary, that Y is Lebesgue measurable. Then X(Y) > 0 and we can find an uncountable set Z c Y of Lebesgue measure zero. But then the set X n Z is uncountable, so we get a contradiction with the definition of the Sierpiliski set X. The contradiction obtained ends the proof of Theorem 5.
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Exercise 14. Let X be a Sierpiriski subset of R considered as a topological space with the induced topology. Show that any Borel subset of X is simultaneously an F,-set and a G6-set in X . In particular, each countable subset of X is a G6-set in X . Exercise 15. Let X be a Sierpiliski subset of R.As mentioned above, all countable subsets of X are Ga-sets in X . Applying this fact, demonstrate that, for any nonempty perfect set P E R , the set X n P is of first category in P. This result strengthenes the corresponding part of Theorem 5. Exercise 16. Let X be a Sierpiliski set on the real line R . Equip X with the topology induced by the density topology of R . Prove that the topological space X is nonseparable and hereditarily Lindelof (the latter means that each subspace of X is Lindelof, i.e, any open covering of a subspace contains a countable subcovering). Exercise 17. Assume that the Continuum Hypothesis holds. Let X be a Sierpiriski set on the real line R.Equip X with the topology induced by the Euclidean topology of R.Prove that
where A(X) denotes the class of all analytic subsets of X and B ( X ) denotes the class of all Borel subsets of X . Exercise 18. Let J1 and J2 be two orthogonal a-ideals of subsets of R, each of which is invariant with respect to the group of all translations of R. Let A1 and A2 be two subsets of R satisfying the relations
Demonstrate that: 1) there exists a set B1 E J1 for which we have
B1 - A2 = U { B 1- a : a E A 2 ) = R; 2) there exists a set B2 E J2 for which we have
B2 - A1 = U { B 2- a : a E A 1 ) = R. Further, put: J1 = the a-ideal of all first category subsets of R ; Jz = the a-ideal of all Lebesgue measure zero subsets of R. Deduce from relations 1) and 2) that if X is an arbitrary Luzin set on R and Y is an arbitrary Sierpiriski set on R, then the equalities
are fulfilled. We thus conclude that the simultaneous existence in R of Luzin and Sierpiriski sets immediately implies that the cardinality of these sets is as minimal as possible (i.e, is equal to the smallest uncountable cardinal). This result wae obtained by Rothberger (see [124]). It is easy to observe that if we replace the Continuum Hypothesis by Martin's Axiom (which is a much weaker assertion than CH), then we can prove the existence of some analogues of Luzin and Sierpiriski sets. Namely, if Martin's Axiom holds, then there exists a set X C R such that: 1) ccard(X) = c; 2) for each set Y E I'(R),we have
A set X with the above property is usually called a generalized Luzin subset of the real line. Similarly, if Martin's Axiom holds, then there exists a set X c R such that: (1) card(X) = c; (2) for each set Z E I ( X ) , we have
A set X with the above property is usually called a generalized Sierpiriski subset of the real line. Let us remark that, for the existence of generalized Luzin sets or generalized Sierpiriski sets, we do not need the full power of Martin's Axiom. In fact, the existence of generalized Luzin and Sierpiliski sets is implied by some additional set-theoretical assumptions which are much weaker than Martin's Axiom. The next exercise contains the corresponding result for an abstract a-ideal of sets.
>
Exercise 19. Let E be a set with card(E) w and let J be a proper a-ideal of subsets of E, containing in itself all one-element subsets of E. We denote: cov(J) = the smallest cardinality of a covering of E by sets belonging to J; cof(J) = the smallest cardinality of a base of J. Prove that if the equalities cov(J) = cof (J) = ccard(E) are fulfilled, then there exists a subset D of E such that ccard(D) = card(E)
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8
and, for any set Z E J, we have
In particular, if our basic set E coincides with the real line R and J is the a-ideal of all first category subsets of R (respectively, the a-ideal of all Lebesgue measure zero subsets of R), then we obtain, under Martin's Axiom, the existence of a generalized Luzin subset of R (respectively, the existence of a generalized Sierpiliski subset of R). Some facts about generalized Luzin sets and generalized Sierpiriski sets are presented in the next three exercises. Exercise 20. Assume that the Continuum Hypothesis holds. Prove that there exists a set X c R satisfying the following conditions: a) X is a vector space over the field Q; b) X is an everywhere dense Luzin subset of R. Show also that there exists a set Y c R satisfying the following conditions: (a) Y is a vector space over the field Q ; (b) Y is an everywhere dense Sierpiliski subset of R . Moreover, by assuming Martin's Axiom, formulate and prove analogous results for generalized Luzin sets and for generalized Sierpiriski sets. In addition, infer from these results, by assuming Martin's Axiom again, that there exist an isomorphism f of the additive group of R onto itself and a generalized Luzin set X in R such that f (X) is a generalized Sierpiriski set in R. Exercise 21. Suppose that Martin's Axiom holds. Prove that any generalized Luzin subset of R is universal measure zero. In addition, by using a generalized Luzin set on R, show that there exists a a-algebra S of subsets of R, such that: a) for each point x E R, we have {x} E S; b) S is a countably generated a-algebra, i.e. there exists a countable subfamily of S which generates S; c) there is no nonzero a-finite diffused measure defined on S. In particular, we see that Martin's Axiom implies that the cardinal c is not real-valued measurable.
A result similar to the one presented in Exercise 21 can be proved in the theory ZFC if we replace R by a certain uncountable subspace E of R. Namely, it suffices to take as E a universal measure zero subset of R with cardinality equal to w l . In particular, we immediately obtain from this result that w l is not real-valued measurable (cf. [161]).
Exercise 22. Assume Martin's Axiom. By applying a generalized Luzin set, prove that there exist two a-algebras S1and S2of subsets of the real line R, satisfying the following conditions: 1) B(R) E Sl ~ S Z ; 2) both S1 and S2are countably generated a-algebras; 3) there exists a measure p1 on S1extending the standard Borel measure on R ; 4) there exists a measure p2 on Sz extending the standard Borel measure on R ; 5) there is no nonzero a-finite diffused measure defined on the a-algebra of sets, generated by S1 U Sz. We wish to present here one application of a generalized Luzin set to the construction of a function which is extremely bad from the point of view of measure theory. First, we must give the corresponding definition. Let E be a set (in particular, a topological space) and let f be a function acting from E into R. We shall say that f is absolutely nonmeasurable if, for any nonzero a-finite diffused measure p on E, our f is nonmeasurable with respect to p. Let us stress that, in this definition, the domain of p is not a fixed aalgebra of subsets of E (actually, d o r n ( ~ )may be an arbitrary a-algebra of subsets of E , containing all singletones). T h e o r e m 6. Suppose that Marian's Axiom holds. Then there exists an injective functaon f : R-+R
whach samultlaneously is absolulely nonmeasurable. Proof. We know that Martin's Axiom implies the existence of a generalized Luzin subset of R . Let X be such a subset. Since we have
there exists a bijection
f : R-+X. Obviously, we can consider f as an injection from R into itself. Let us verify that f is the required function. Suppose, for a moment, that our f is not absolutely nonmeasurable. Then there exists a nonzero a-finite diffused measure p on R such that f is p-measurable, i.e. for any Borel subset B of R, the relation f - l ( ~ E) d o m b )
is satisfied. Equivalently, for any Borel subset B' of X , we have f-'(8') E dom(p). Clearly, without loss of generality, we may assume that p is a probability measure. Now, for each Borel subset B1 of X , we put
In this way we obtain a Borel diffused probability measure v on X , which is impossible since X is a universal measure zero space (see Exercise 21 above). This contradiction ends the proof of Theorem 6.
Remark 3. The preceding theorem was formulated and proved under the assumption that Martin's Axiom is valid. In this connection, it is reasonable to point out here that we cannot establish the existence of an absolutely nonmeasurable function within the theory ZFC.Indeed, if the cardinality of the continuum is real-valued measurable, then such functions do not exist. At the same time, one can easily demonstrate (in ZFC) that there exists an absolutely nonmeasurable function In order to show this, it suffices to pick a universal measure zero subspace X of R with card(X) = w l and then to take as f any bijection acting from w l onto X . In our further considerations, we shall meet some other applications of Luzin sets and Sierpifiski sets (respectively, of generalized Luzin sets and generalized Sierpiriski sets). But now we shall use once more Martin's Axiom and construct a generalized Sierpiriski set with the Baire property in the restricted sense.
Theorem 7. Suppose that Madin's Axiom holds. Then there exists a set X c R such that: 1) for every nonempty perfect set P E R, the intersection X n P is a first category set in P; 2) for each Lebesgue measurable set Y E R with X(Y) > 0, the inter'section X n Y is nonempty; 3) X is a generalized Sierpin'ski subset of R.
Proof. Let, as usual, c denote the cardinality of the continuum. Obviously, we may identify c with the smallest ordinal number a whose cardinality is equal to c. Let (Z()(<, denote the family of all Borel subsets of the real line with a strictly positive Lebesgue measure, i.e.
and let (TS),<, denote the family of all Bore1 subsets of the real line having Lebesgue measure zero, i.e.
For each ordinal 6 < a , we fix a partition {Z:, 2:) of 2, such that 2: is a Lebesgue measure zero set and 2; is a first category subset of Z t . Notice that the existence of such a partition follows directly from Lemma 1. Now, we define an injective a-sequence (x,),<, of real numbers, Suppose that 6 < a and that the partial sequence ( x ~ ) ~has < , already been defined. Let us consider the set
4 =(U~ , O ) U { X C (5,
:
c
Martin's Axiom implies that the set D, is also of Lebesgue measure zero. Hence we have 2, \De # 0So we can choose a point X E from the last nonempty difference of sets. In this way, we are able to define the whole a-sequence (xf),<, of points of R. Now, we put x = { x , : 6 < a) and we are going to show that the set X is the required one. Let P be any nonempty perfect subset of R. If its Lebesgue measure is equal to zero, then, for some ordinal 6 < a, we may write P = T E . Consequently, from the method of construction of the set X , we immediately obtain
Applying Martin's Axiom again, we see that the intersection P n X is a first category set in P. Suppose now that X(P) > 0. Then, for some ordinal < a, we can write P = Z E . Therefore
Taking account of the fact that the set P does not have isolated points, we obtain from the last inclusion that PnX is a first category set in P. Hence condition 1) is satisfied for our set X . Furthermore, since
for each ordinal 5 < a, we conclude that condition 2) holds for the set X , too. The validity of condition 3) follows directly from our construction.
Theorem 7 has thus been proved. R e m a r k 4. This result enables us to conclude that Martin's Axiom implies the existence of a generalized Sierpiriski subset of R which is thick (with respect to the Lebesgue measure A) and simultaneously has the Baire property in the restricted sense. We want to finish this chapter with some facts and statements concerning universal measure zero sets and strongly measure zero sets. Exercise 23. Check that if A is an arbitrary first category subset of R , then the equality (R\A)+(R\A)= R holds true. By starting with this fact, assuming the Continuum Hypothesis and using the method of transfinite recursion, construct a Luzin set X on R such that X+X=R. Formulate and prove an analogous result (under Martin's Axiom) for a generalized Luzin set. Exercise 24. Let Z be a subset of the Euclidean plane R 2 . We say that Z is strongly measure zero if, for each sequence { E , : n E w ) of strictly positive real numbers, there exists a countable covering {V, : n E w ) of Z by squares, such that (Vn E w)(diam(V,)
< s,).
Let now I be a straight line in R~ not parallel to the line R x (0). Consider the projection P'(R,,) : R2 3 R of R 2 onto R , in the direction I . Show that if Z is a strongly measure zero subset of R 2 , then P P ( ~ , ! ) ( Zis) a strongly measure zero subset of R . In addition, introduce the notion of a Luzin subset (respectively, of a generalized Luzin subset) of the plane R 2 and prove that, under the Continuum Hypothesis (respectively, under Martin's Axiom), there exist Luzin subsets (respectively, generalized Luzin subsets) of R 2 . Finally, verify that any Luzin set in R 2 is strongly measure zero (and hence universal measure zero). Exercise 25. Let X denote the Luzin set of Exercise 23 and let
be a mapping defined by the formula
Check that
4 coincides with the
projection p r ( ~ , of , ) R 2 onto
R,where
Show also that 4 ( X x X ) = R. The latter property of X implies, in particular, that the product X x X is not a strongly measure zero subset of R 2 (cf, the result of Exercise 24). Conclude from this fact that the Cartesian product of two strongly measure zero subsets of R is not, in general, a strongly measure zero subset of R2.In this connection, let us recall that the class of all universal measure zero spaces is closed under finite Cartesian products (see Exercise 4 of the present chapter). Note that the results given in Exercises 23 Sierpiliski (cf. [140]).
-
25 are essentially due to
The following statement (due to Marczewski) yields a characterization of universal measure zero subsets of R in terms of Lebesgue measure zero sets.
Theorem 8. Let X be a subset of R. Then these two assertions are equivalent: 1) X is a universal measure zero space; 2) each homeomorphac image of X lying in R has Lebesgue measure zero. Proof. Suppose first that X satisfies relation 1). Let Y be a subset of
R homeomorphic to X . Fix any homeomorphism If X*(Y)> 0, then obviously there exists a nonzero a-finite diffused Borel measure p on Y. Putting
we obtain a nonzero a-finite diffused Borel measure v on X . But this is impossible because X is a universal measure zero space. Implication 1) =+ 2) has thus been proved.
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Let now X satisfy relation 2 ) . We are going to demonstrate that relation 1) holds for X , too. Of course, without loss of generality, we may assume that our X is a subset of the unit segment [O,l]. Suppose for a moment that X is not a universal measure zero space. Then there exists a Borel diffused probability measure p on [O,1]such that
We may also assume that p does not vanish on any nonempty open subinterval of [O,1] (replacing, if necessary, p by (p X)/2). Now, define a function f : [O,ll -+[O,lI
+
by the formula
f ( 4 = P([0>4 )
(a: E [0,11).
Evidently, f is an increasing homeomorphism from [O,l] onto itself. Let us put 4 2 ) = ~ (- l (fz ) ) (2 E B([O, 11)). In this way we get a Borel probability measure v on [0, 11 such that
Furthermore, it turns out that v coincides with the standard Borel measure on [0, 11. Indeed, for each interval [a, b] C_ [0, 11, we may write
where ~ ( [ 0cl) , = a,
cl([O,dl) = b .
Then we have .([a, bl) = P ( f -l([a, bl)) = P([c,
4)=
= cl([0,4)- ~ ( [ 0cl) 1 = b - aConsequently, the measures v and X are identical on the family of all subintervals of [0, 11 and hence these two measures coincide on the whole Borel a-algebra of [ O , l ] . Thus
which contradicts relation 2 ) . The contradiction obtained establishes implication 2 ) 3 1) and ends the proof of Theorem 8.
Exercise 26. Let X be an uncountable topological space such that all one-element subsets of X are Borel in X . We say that X is a Sierpiliski space if X contains no universal measure zero subspace with cardinality equal to card(X). Show that: a) any generalized Sierpiliski subset of R is a Sierpiriski space; b) if X is a Sierpiliski space of cardinality w l , then A(X) = B(X), where A(X) denotes the class of all analytic subsets of X (i.e, the class of all those sets which can be obtained by applying the (A)-operation to various (A)-systems consisting of Borel subsets of X ) and B(X) denotes, as usual, the class of all Borel subsets of X ; c) if X1 and X2 are two Sierpiliski spaces and X is their topological sum, then X is a Sierpiriski space, too; d) if X is a Sierpiriski space, Y is a topological space such that card(Y) = car$(X), all one-element subsets of Y are Borel in Y , and there exists a Borel surjection from X onto Y, then Y is a Sierpiliski space, too. Consequently, if X is a Sierpiriski subset of R and
is a Borel mapping such that card(f(X)) = card(X), then f ( X ) is a Sierpiliski subspace of R . Exercise 27. By assuming Martin's Axiom and applying the method of transfinite recursion, construct a generalized Sierpiriski subset X of R such that X+X=R. Infer from this equality that there exists a continuous surjection from the product space X x X onto R. Further, by starting with this property of X and taking into account assertion d) of the preceding exercise, show that the product X x X is not a Sierpiriski space. Conclude from this fact that the topological product of two Sierpiliski spaces is not, in general, a Sierpiliski space. Exercise 28. Let H be a Hilbert space (over the field R ) whose Hilbert dimension is equal to c (in particular, the cardinality of H equals c, too). Assuming that c is not a real-valued measurable cardinal, demonstrate that there exists a subset X of H satisfying the following conditions: 1) car$(X) = c; 2) X is everywhere dense in H (in particular, X is nonseparable); 3) X is a universal measure zero subspace of H.
Suppose now that c is not cofinal with w l , i.e, c cannot be represented in the form c= IC(
C
€<w1
where all cardinal numbers I E ~(< < wl) are strictly less than c . By starting with the fact that there exists an wl-sequence of nowhere dense subsets of H covering H, show that there is no generalized Luzin subset of H.In other words, show that there is no subset Y of H satisfying the next two relations: a) ccard(Y) = c; b) for each first category set Z C H ,the inequality
is fulfilled. Rich additional information about Luzin sets and Sierpiliski sets (also, about other small subsets of the real line) can be found in [89] and [107].
9. Egorov type theorems
It is well known that one of the earliest important results in real analysis and Lebesgue measure theory was obtained by Egorov [37] who discovered close relationships between the uniform convergence and the convergence almost everywhere of a sequence of real-valued Lebesgue measurable functions. This classical result (known now as the Egorov theorem) has numerous consequences and applications in analysis. For example, it suffices to mention that another classical result in real analysis - the so-called Luzin theorem on the structure of Lebesgue measurable functions - can easily be deduced by starting with the Egorov theorem. Here we wish to discuss some aspects of the Egorov theorem and, in addition, to show that, for a sequence of arbitrary real-valued functions there is no hope to get a reasonable analogue of this theorem. In other words, we are going to demonstrate in our further considerations that there are some sequences of rather strange real-valued functions, for which even very weak analogues of the Egorov type theorems fail to be true. First of all, we want to present the Egorov theorem in a form slightly more general than those in which this theorem is usually formulated in various standard courses of real analysis and classical measure theory. In order to do this, we need some auxiliary notions and facts. Let E be a nonempty set and let S be some class of subsets of E, satisfying the following conditions: 1) 0 E S, E E S; 2) S is closed under countable unions and countable intersections. Suppose also that a functional
is given, such that: a) for every increasing (with respect to inclusion) sequence of sets
we have U(u{xn :
72
< w)) < ~ u P { v ( X ~: )71 < w ) ;
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b) for every decreasing (with respect to inclusion) sequence of sets
we have v(fl{Yn : n
< w)) 2 inf{v(Yn)
: n
< w).
In this case, we say that S is an admissible class of subsets of E and v is an admissible functional on S. Note that, in analysis, there are many natural examples of admissible functionals. This can be confirmed by the following example. E x a m p l e 1. Let E be a nonempty set and let S be some a-algebra of subsets of E . Then it is obvious that S is an admissible class. Suppose, in addition, that v is a finite measure on S. Then it can easily be observed that v is an admissible functional on E . Exercise 1. Give an example of an admissible functional on E, which is not a measure on E. Similarly to the concept of measurability of real-valued functions with respect to ordinary measures, the concept of measurability of real-valued functions with respect to admissible functionals may be introduced and investigated. Namely, we say that a function
is measurable with respect to an admissible functional v on E (or, simply, f is Y-measurable) if, for each open interval It, t l [ c R, we have
Obviously, the same definition can be introduced for partial functions acting from E into R. The properties of functions (partial functions) measurable with respect to admissible functionals turn out to be rather similar to the properties of functions (partial functions) measurable in the usual sense. The next two exercises vividly illustrate this fact. Exercise 2. Let v be an admissible functional on ,E and let
+
be any two v-measurable functions. Show that the functions f g and f - g are v-measurable, too. In addition, show that if g(x) # 0 for all x E E, then the function f/g is also v-measurable.
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183
Let X be an arbitrary set from d o m ( v ) . Check that the restriction f (X is a v-measurable partial function from E into R.
Exercise 3. Let v be an admissible functional given on a set E and let {fn : n < w ) be a sequence of v-measurable functions, pointwise convergent on E . Let us denote
Demonstrate that the function f is measurable with respect to v . Now, we are able to formulate and prove a direct analogue of the Egorov theorem for sequences of real-valued functions measurable with respect to an admissible functional.
Theorem 1. Let E be a basic set and let v be an admissible functional on E . Suppose, in addition, that a sequence {fn : n < w ) of v-measurable functions as given, pointwise convergent on E l and let f denote the corresponding limit function. Then, for each E > 0, there exasts a set X E d o m ( v ) satisfying these two relations: 1 ) v ( E ) - v(X) _< E ; 2) the sequence of functions { f n ( X : n < w ) converges uniformly to the function f J X . Proof. For any natural number m , let us denote
It is easy to check that the set E o l mbelongs to d o m ( v ) ,and
Consequently, there exists a natural index mo such that
Let us put Xo = Eo,m, and, for each m < w , consider the set
Evidently, E l , , belongs to d o m ( v ) , and
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Consequently, there exists a natural index m l such that
Let us put X1 = El,,, . Continuing in this manner, we will be able to define a certain sequence { X I , : k < w ) of sets belonging to d o m ( v ) and satisfying the relations: a) Xo 2 X I 2 ... 2 X k ,.,; b) for any k < w , we have v(E) u ( X l c )< E ; c) for any k < w , there is a natural number mr, for which we have
>
Finally, we put
-
x =nixk :
k <w).
Then, in virtue of the definition of an admissible functional, we get
and it can easily be verified that the sequence of the restricted functions { f n l X : n < w ) converges uniformly to the restricted function f ( X . This completes the proof of Theorem 1. Obviously, the theorem presented above immediately implies the classical Egorov theorem (see [37]). It suffices to take as v an arbitrary finite measure on E . In this connection, let us recall that, for a-finite measures, a direct analogue of the Egorov theorem is not true in general. Exercise 4. Let X denote, as usual, the standard Lebesgue measure on the real line R. Give an example of a sequence { f , : n < w ) of real-valued uniformly bounded A-measurable (even continuous) functions on R , which is convergent everywhere on R but there exists no unbounded subset X of R such that the sequence { f n l X : n < w) is uniformly convergent on X . Exercise 5. Let E be a normal topological space and let p be a finite inner regular Borel measure on E , i.e. for each Borel subset Y of E, we have the equality p(Y) = sup{p(F) : F
Y & F is closed in E ) .
We denote by p' the usual completion of p . Let
E G O R O V T Y PE THE O REM S
185
be an arbitrary pl-measurable function. By applying the Tietze-Urysohn theorem on the existence of a continuous extension of a continuous realvalued function defined on any closed subset of E , show that, for each E > 0, there exists a continuous function
satisfying the relation
Deduce from this fact that, for any pl-measurable function
there exists a sequence (4, : n < w ) of continuous real-valued functions on E, convergent to 4 almost everywhere (with respect to pl). Exercise 6. Let E be again a normal topological space, let p be a finite inner regular Borel measure on E and let p1 denote the completion of p. By starting with the Egorov theorem and applying the result of Exercise 5, prove the following Luzin type theorem: for any pl-measurable function
and for each real
E
> 0, there exists a continuous function
such that PI({. E E : f ( x ) # g(z))) < E The latter relation just expresses that the given function f has the so-called (C)-property of Luzin. It is frequently said that all pl-measurable functions possess this property (and the converse assertion is true, too). Let us now return to the Egorov theorem. In conformity with it, any convergent sequence of measurable real-valued functions converges uniformly on some "large" measurable subset of E ("large" means here that the measure of the complement of this subset may be taken arbitrarily small). In particular, if a given finite Borel measure on E is nonzero diffused and inner regular, then we immediately obtain that every convergent sequence of measurable real-valued functions on E converges uniformly on an uncountable closed subset of E. Hence, if E is an uncountable Polish topological space equipped with a nonzero finite diffused Borel measure, then, for any
186
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convergent sequence of measurable real-valued functions on E , there exists a nonempty compact perfect subset of E (actually, a subset homeomorphic to the Cantor discontinuum) on which the sequence converges uniformly. In connection with these observations, it makes sense to consider the following more general situation. Let E be an arbitrary uncountable complete metric space without isolated points and let { f , : n < w ) be a sequence of real-valued Borel functions on E , such that, for some constant d 2 0, we have
in other words, our sequence of functions is uniformly bounded. Do there exist a nonempty perfect compact subset P of E and an infinite subset K of w , for which the partial sequence of functions { f n l P : n E I<} is uniformly convergent on P? Evidently, we may restrict our considerations to the case where the given space E is homeomorphic to the Cantor discontinuum (since, according to the well-known theorem from general topology, every complete metric space without isolated points contains a homeomorphic image of the Cantor discontinuum). Actually, we may suppose from the beginning that our original space E is uncountable and Polish. Also, in order to get a positive solution of the question formulated above, it suffices to demonstrate that there exist an infinite subset Ii' of w and a nonempty perfect compact subset P' of E , such that the sequence
converges pointwise on P'. Indeed,. suppose that this fact has already been established and equip the set P' with some Borel diffused probability measure p. Then, evidently, we may apply the Egorov theorem to p and { f n ( P ' : n E I<). In conformity with this theorem, there exists a Borel set X C P' with p ( X ) > 112,for which the sequence of functions { f n l X : n E I<) converges uniformly. Since p is diffused and p ( X ) > 0, we obviously obtain the relation
and, consequently,
card(X) = c because X is Borel in P'. It is clear now that X contains a nonempty perfect compact subset P for which the sequence of functions { f n l P : n E Ii'} converges uniformly, too.
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E G O R O V T Y PE THE O REM S
Mazurkiewicz was the first mathematician to prove that, for any uniformly bounded sequence of real-valued Borel functions given on an uncountable Polish space E, there exists a subset of E homeomorphic to the Cantor discontinuum, on which some subsequence of the sequence converges uniformly. In order to present a detailed proof of this interesting result, we need some auxiliary notions and simple facts. Let E be an uncountable Polish space and let be a family of realvalued functions defined on E . We shall say that the family is semicompact if, for each sequence { : n < w) E and for each nonempty perfect set P E, there exist an infinite subset IC of w and a nonempty perfect set PI contained in P , such that the partial sequence of functions (4, : n E IC) converges pointwise on PI. We shall say that a family S consisting of some Borel subsets of E is semicompact if the corresponding family of characteristic functions
is semicompact in the sense of the definition above. The following auxiliary statement yields a much more vivid description of semicompact families of Borel sets in E.
Lemma 1. Let S be a family of Borel subsets of an uncountable Polish space E. Then these two asseriions are equivalent: 1) the family S is semicompaci; 2) for any sequence {X, : n < w ) of sets from S and for each nonempty perfect subset P of E , there exists an infinite set I< E w such that
Proof. Note first that implication 2) for example, we have
a 1) is almost
trivial because if,
for some infinite subset K of w, then the set
n{Xn
: n E IC) fl P
contains a noneinpty perfect subset PI, and the sequence of characteristic functions {xx, : n E Ii') converges pointwise to the function xp, on the set PI (since all xx, (n E Ii') are identically equal to 1 on P').
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Now, let us establish implication 1) j 2). Suppose that relation 1) is fulfilled. Let { X , : n < w ) be an arbitrar y sequence of sets from S and let P be a nonempty perfect subset of E. We may assume, without loss of generality, that P = E. Then, according to I ) , there exists an infinite subset Ii' of w such that the corresponding sequence of characteristic functions { x x , : n E Ii') is convergent on an uncountable Borel subset Y of E. Let us denote
Obviously, x is a Borel function on Y, and
Therefore, at least one of the sets
is uncountatile. Suppose, for example, that card(Yl) > w . Then, by taking account of the formula Yl = Yl r l limsup{Xn : n E IC) = Yl r l l i m i nf { X , : n E I f ) , it can easily be checked that, for some infinite subset
of Ii', the inequality
is satisfied. If card(Yo) > w , then an analogous argument applied to the sequence of characteristic functions
yields the existence of an infinite subset KO of IC for which the inequality
is fulfilled. This establishes implication 1) Lemma 1.
j
2) and finishes the proof of
The next two auxiliary propositions also are not hard t o prove. Lemma 2. Let be a semicompact family of real-valued functions defined on an uncountable Polish space E . Then, for any real number d 2 0, the famaly offundaons
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E G O R O V T Y PE THE O R EM S
is semicompact, too. Lemma 3. Let a1 and be any two semicompact families of realvalued functions defined on an uncountable Polish space E . Then the family of functions {41+42 :
41 € @ I , 4 2 € @ 2 )
is semicompact, too. Exercise 7. Give the detailed proofs of Lemma 2 and Lemma 3. It immediately follows from these lemmas (by using the method of induction) that if d 1 0 and alla2,.,., akare semicompact families of functions on an uncountable Polish space E , then the family of all those functions which can be represented in the form
where It11
41
5 d, It21 5 d, .,. , ltkl 5 d, 42
E
. a -
4 k E ahl
is also semicompact.
Lemma 4. Let be a semicompact family of bounded real-va/ued functaons on an uncountable Polish space E , and let a* denote the family of all those functions which are uniform limits of sequences of funclaons belonging to (in other words, a* is the closure of with respect to the topology of una'form convergence on E). Then the family a* is semicompact, too. Proof. Let (4: : n < w ) be an arbitrary sequence of functions from the family a*.In virtue of the definition of a*,for every natural number n , there exists a function 4, E such that
Let us consider the family of functions (4, : n < w ) . According to our assumption, is semicompact. Hence, for each nonempty perfect set P 2 E l there exist an infinite subset Ii' of w and a nonempty perfect subset P' of P, such that the partial sequence of functions
{dn1P1 : n E I<} converges pointwise on P I to some function 4 defined on Pi. Now, it is easily verified that the corresponding sequence of functions
190
also converges pointwise to
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9
4. This completes the proof of Lemma 4.
The following auxiliary statement plays the key role in our further considerations.
Lemma 5. The family of all Borel subsets of an uncountable Polish topological space E zs semicompact.
Proof. Let { X , : n < w ) be an arbitrary sequence of Borel subsets of E and let P be a nonempty perfect set in E. We are going to construct (by recursion) a dyadic family
of nonempty perfect sets in E and a sequence {nk : k numbers, such that: a) P0 is contained in P ; b) for any a E 2 < W , we have
< w ) of natural
c) for any nonempty a E 2 < W , we have
where t h e symbol lh(a) denotes the length of a ; d) the sequence {nk : k < w ) is strictly increasing; e) for each nonzero k < w , the inclusion
is fulfilled. Pick a nonempty perfect set P0 C_ P and a natural number no arbitrarily. Suppose now that the natural numbers
and the partial family
of nonempty perfect sets in E have already been defined. Only two cases are possible. 1. There exists a natural number n > nk for which all the sets
19 1
E G O R O V T Y PE T H E O R EM S
are uncountable. In this case, we may put nk+l = n and, for any a E 2<W with Ih(a) = k, we can construct two nonempty perfect sets POoand Pal satisfying the relations
So we see that, in this case, the process of our construction can be continued. 2 . For each natural number n > nk, there exists a u from 2<W with Ih(a) = k, such that card(P, n X,) 5 w . Since the family {Po : Ih(u) = k) is finite, we can find an infinite subset M of w and a a / E 2<W with lh(al)= k, such that
card(P,~n X,) 5 w for all numbers n belonging to M . From the latter relation we obtain
which immediately gives the desired result. Thus we may restrict our considerations only to case 1. As mentioned above, in this case, the construction described can be continued and, after w many steps, yields a dyadic family
of nonempty perfect subsets of E. Now, putting
P' =
(U{P, : Ih(u) = k)),
we get a nonempty perfect set P' such that
Hence, we may write
and the proof of Lemma 5 is complete. Now, taking into account the preceding lemmas, we are able to formulate and prove the following result of Mazurkiewicz.
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T h e o r e m 2. Let be an arbatrary uniformly bounded famaly of realvalued Borel functions on an uncountable Polish space E. Then @ is semicompact. Proof. Since our @ is uniformly bounded, there exists a real number d 2 0 such that (Vd E @)(IldJIIL 4. Let us denote by \E' the family of all those functions II, which satisfy the following two relations: 1) IIII,II L d ; 2 ) II, is a linear combination of characteristic functions of some Borel subsets of E . Then, according to Lemmas 2, 3 and 5, the family \E' is semicompact. Also, it is clear that the original family @ is contained in the closure of \E' (with respect to the topology of uniform convergence on E). Hence, in view of Lemma 4, the family is semicompact, too. This ends the proof of Mazurkiewicz's theorem. Let us observe that if E = R, then Theorem 2 can be extended to an arbitrary family of uniformly bounded real-valued Lebesgue measurable functions on E and to an arbitrary family of uniformly bounded real-valued functions on E having the Baire property. Indeed, in order to obtain the corresponding results, it suffices to apply the following well-known fact: for any Lebesgue measurable (respectively, having the Baire property) realvalued function on R, there exists a real-valued Borel function II, on R such that the set 1. E R : dJ(4 # II,(x))
+
is of Lebesgue measure zero (respectively, of first category)< Exercise 8. Let E be an uncountable Polish topological space. Prove an analogue of Theorem 2 for real-valued functions on E possessing the Baire property and for real-valued functions on E measurable with respect to the completion of a fixed nonzero a-finite diffused Borel measure on E. In addition, give an example of a sequence {f, : n < w ) of uniformly bounded real-valued Borel functions on R, such that, for each infinite subset M of w , the corresponding partial sequence {f, : n E M ) is convergent only on a first category subset of R being simultaneously of Lebesgue measure zero. As demonstrated above, for any uniformly bounded sequence of realvalued functions possessing good descriptive properties, we have the pointwise convergence (and even the uniform convergence) of an appropriate
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E G O R O V T Y PE TH E O REM S
subsequence on some nonempty perfect set, hence, on some set of cardinality continuum. However, various uniformly bounded sequences of realvalued functions are possible, which are extremely bad from the point of view of convergence pointwise. The following statement (essentially due to Sierpiriski) shows that the existence of such sequences can be directly deduced from the existence of a Luzin set Z in an uncountable Polish space E with card(Z) = c . In this connection, it is reasonable to recall here that the existence of a Luzin set of cardinality continuum is easily implied by the Continuum Hypothesis (see, e.g., Chapter 8 of the book).
Theorem 3. Let Z be a Luzin subset of the Cantor discontinuurn 2", satisfying the equality card(Z) = c . Then there exists a sequence {X, : n < w ) of subsets of 2") such that, for each infinite subset I< of w, the corresponding partial sequence of characteristic functions {xx, : n E I<) converges pointwise only on a countable subset of 2".
Proof. For every natural number n, let us denote
The sets B, (n < w) and their complements are clopen in 2" and generate a base of the standard product topology on 2". It can easily be seen that, for any infinite subset Ii' of w , the intersections
are nowhere dense closed subsets of 2". Since Z is a Luzin set in 2") we have card(n{B, : n E Ii') n Z ) 5 w , card(n(2"
\ Bn
: n E Ii')
n Z) 5 w .
Now, let
h : 2"
-*
2"
be an injection such that h(2") = Z. We put X, = h - ' ( ~ , )
(n
< w).
CHAPTER
9
Consider the sequence of characteristic functions
We assert that this sequence is the required one. Indeed, it immediately follows from the definition of the family of sets {Xn : n < w ) that, for any infinite subset h' of w , the intersections
are at most countable. But from this fact we easily infer that the corresponding partial sequence of characteristic functions {xxn: n E h') can be convergent pointwise only on a countable subset of 2W.Theorem 3 has thus been proved. Exercise 9. Formulate and prove an analogue of Theorem 3 in the situation when there exists a generalized Luzin subset of 2W.We recall that the existence of generalized Luzin sets follows, for instance, from Martin's Axiom (see Chapter 8). In addition, formulate and prove two statements analogous to Theorem 3 under the assumption of the existence of a Sierpiriski set in 2?(respectively, of a generalized Sierpiiski set in 2W). The two results above (namely, Theorem 3 and the corresponding result of Exercise 9) were established under the assumption of the existence of a Luzin set (generalized Luzin set) with cardinality equal to c. As we know (see, e.g., Chapter 8), if Martin's Axiom and the negation of the Continuum Hypothesis hold, then there are no Luzin sets on the real line R (and, hence, in the Cantor discontinuum 2W). So, in such a case, the above argument does not work. The last theorem of this chapter, presented below, shows us that, under Martin's Axiom, the pointwise convergence of an appropriate subsequence of real-valued functions can be achieved for any subset with small cardinality (i.e. with cardinality strictly less than c). Actually, in order to establish the desired result, we do not need the full power of Martin's Axiom. It suffices to apply one purely combinatorial assertion concerning certain families of infinite subsets of w. This combinatorial assertion is presented in the following exercise. Exercise 10. For any two subsets M and Ii' of w , we shall write if card(M \ I<) < w. Obviously, the relation 5 is a preordering on the family of all subsets of w.
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EQOROV T Y P E THEOREMS
Suppose that Martin's Axiom holds. Let K be an infinite cardinal strictly less than c (as usual, we identify K with the smallest ordinal number having the same cardinality), and let {Mt : ( < K ) be a family of infinite sets in w , such that (VO(VC)(E 5 C < fc Mc 5 ME).
*
Prove that there exists an infinite subset M of w satisfying the relation
In addition, show in the theory infinite subsets of w , such that
ZF that if {M,
: n
<w )
is a sequence of
then there exists an infinite set M C w satisfying the relation
Some other combinatorial consequences of Martin's Axiom closely related to the one presented in Exercise 10 are discussed in [55] and [88].
Theorem 4. Suppose that Martin's Axiom holds. Let E be an arbitrary set of cardinality continuum, let X be a subset of E with
and let {fn : n < w ) be a uniformly bounded sequence of real-valued functaons given on E . Then there exasts an infinite subset M of w such that the partial sequence of functions {fnlX : n E M ) is convergent pointwise on X .
Proof. Let us put
fc
= c a r d ( X ) and let
be some enumeration of all elements of X . By applying the result of Exercise 10, it is not hard to define recursively a family {Me : ( < K ) of infinite subsets of w , satisfying the following conditions: a) Mc 5 Me for ( -< C < f c ; b) for each ( < K , the partial sequence of real numbers
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C H APTER 9
is convergent. Now, applying the result of Exercise 10 once more, we can define an infinite subset M of w such that
Then it is easily verified that the partial sequence of functions
converges pointwise on the whole set X , This ends the proof of Theorem 4. In addition, we wish to remark that the method just described turns out to be rather useful in those questions of analysis which are concerned with various kinds of convergence of sequences of real-valued functions on a given set E . In fact, this method may be regarded as some generalization of the well-known diagonalization method of Cantor. Notice, finally, that purely combinatorial arguments (similar to the one presented above) have found numerous applications in real analysis (see, for instance, [22] and [28]).
Exercise 11. Let X be a Polish topological space equipped with a Borel probability measure p , let
be a Borel mapping and suppose that, for each x E X , there exists a limit limt,oF(x,t) = f(x). Starting with the fact that the projection of a Borel subset of a Polish product space is an analytic set, prove the following parametric version of the Egorov theorem: for any E > 0, there exists ) f(y) a Borel set Y c X such that p(Y) > 1 - E and l i m t _ , ~ F ( y , t = uniformly with respect to y E Y . This result is due to Tolstov.
10. Sierpiliski's partition of the Euclidean plane
In this chapter, we discuss several results and statements which are tightly connected with the classical Sierpidski partition of the Euclidean plane R~ = R x R. It turns out that these results and statements can be successfully applied in various fields of mathematics. Especially, they can be applied to certain questions and problems from mathematical analysis, measure theory and general topology.
,
Let w denote, as usual, the least infinite ordinal number and let w l denote the least uncountable ordinal number. It is a well-known fact that Sierpidski was the first mathematician who considered, in his classical paper [142],a partition {A, B) of the product set w l x w l , defined as follows:
B = { ( € t o :Wl > E > 0. He observed that, for any E < w l and 5 < w l , the inequalities
are fulfilled, where
= {E : (€56) E A), BE = (6.: ( € , < IE B ) . In other words, each of the sets A and B can be represented as the union of a countable family of "curves" lying in the product set w l x w l . This property of the partition {A, B) implies many interesting and important consequences. For instance, it immediately :follows from the existence of {A, B ) that if the Continuum Hypothesis
holds, then there exists a partition {A', B') of the Euclidean plane R 2 , satisfying the relations:
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C HAPTER 10
1) for each straight line L in R2 parallel to the line R x {O), the inequality card(A1n L) w
<
is fulfilled; 2) for each straight line M in R 2 parallel to the line (0) x R, the inequality card(B 1 n M) w
<
is fulfilled. Moreover, Sierpiriski demonstrated that if a covering {A', B') of R 2 with the above-mentioned properties 1) and 2) does exist, then the Continuum Hypothesis is valid. Indeed, suppose that {A', B') is such a covering of R 2. Choose an arbitrary subset X of R having cardinality w l , and put
Then, according to relation 2), we have
On the other hand, let us show that
In order to do this, take an arbitrary point y E R a n d consider the straight line R x {y). Relation 1) implies that card(A1 n ( R x {y)))
4 w.
At the same time, we obviously have card((X x R ) r l ( R x {y))) = w l . Hence there exists a point t E R such that
Since {A', BI) is a covering of R 2 , we infer that ( t ,y) E B' and, consequently, ( t , ~ E) Z, Y E ~ r 2 ( Z ) , which yields the desired equality prz(Z) = R. We thus get
and, finally, c = w l . In other words, Sierpiiski showed that the Continuum Hypothesis is equivalent to the statement that there exists a partition {A', B') of the Euclidean plane R ~satisfying , relations 1) and 2). Exercise 1. Let El and E2 be any two sets, such that
Check that the following assertions are equivalent: 1) the Continuum Hypothesis; 2) there exists a covering (partition) {A, B) of the product set El x Ez, satisfying the relations:
(Vx E El)(card(B n ({x) x Ez))5 w ) . Let us mention an important consequence of the existence of a Sierpitiski partition {A', B') of R 2 . For this purpose, consider the sets
Then we obtain a partition {A", B") of [O, 112 with the properties very similar to the ones of {A', B'). Let us introduce the characteristic functions
It can easily be observed that there exist the iterated integrals
but we have
CHAPTER
10
and, consequently,
Thus we infer that an analogue of the classical Fubini theorem does not hold for each of the functions f and g which, obviously, are nonnegative and bounded on [O, 112(in particular, both f and g are nonmeasurable in the Lebesgue sense).
Remark 1. We see that the Continuum Hypothesis implies the existence of a function f acting from [O, 112 into [O,1] such that its iterated integrals differ from each other. It is not hard to verify that CH is not necessary for this conclusion. For instance, Martin's Axiom also implies the existence of such a function (and, moreover, we do not need here the whole power of MA, it suffices to assume that each subset of R whose cardinality is strictly less than c is measurable in the Lebesgue sense). On the other hand, it was shown in [48] that there are models of set theory in which, for every function g : [0,112 [0,11, the existence of the iterated integrals
implies their equality. For some further statements concerning iterated integrals and tightly connected with the Sierpihski partition, see [131]. Roughly speaking, we do not have any equivalent of CH in terms of iterated integrals. Below, we shall see that there is a beautiful equivalent of CH in terms of differentiability.
Exercise 2. Define a function
+
putting f (a, y) = xy/(x2 y2)2if (2, y) # (0,0), and f (x, y) = 0 if (a, y) = (0,O). Demonstrate that: a) f is Lebesgue measurable but is not Lebesgue integrable;
b) the iterated integrals for f do exist and
In connection with this exercise, let us note that a much more general result is contained in the old paper by Fichtenholz [44]. We now formulate one statement (also interesting from the point of view of measure theory) which follows from the existence of a Sierpiliski partition { A , B ) of the product set w l x w l . If P ( w l ) is the a-algebra of all subsets of w l , then the product a-algebra P ( w l ) 8 P ( w l ) coincides with the a-algebra P ( w l x w l ) of all subsets of W1 X W1.
In order to establish this result, it is sufficient to consider an arbitrary embedding of wl into the real line R and to apply the well-known fact that the graph of any real-valued measurable function is a measurable subset of the product space.
Exercise 3. Give a detailed proof of the equality
Note that an argument establishing this equality relies essentially on the Axiom of Choice because the existence of an embedding of w l into R cannot be proved in the theory ZF & DC. Moreover, as has been shown by Shelah and Raisonnier (cf. [120]), the existence of such an embedding implies in ZF & DC the existence of a subset of R nonmeasurable in the Lebesgue sense. From the equality P ( w 1 x w l ) = P ( w 1 ) 8 P ( w l ) we can directly obtain the following important statement. There does not exist a nonzero c-finite diffused measure p defined on the a-algebra P ( w l ) . Let us recall that this classical statement is due to Ulam [I611 who established the nonexistence of such a measure in another way, by applying a transfinite matrix of a special type (for details, see e.g. [$$I, [110], [I171 or [161]).
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10
In order to prove this statement by using the corresponding properties of the partition { A , B), suppose for a moment that such a measure LI, does exist and apply the Fubini theorem to the product measure LI, x LI, and to the sets A and B of the Sierpiriski partition. We immediately get the equalities
and, consequently,
which yields a contradiction. Thus w l is not a real-valued measurable cardinal. In fact, the real-valued nonmeasurability of w l is historically the first nontrivial statement which concerns some important combinatorial properties of uncountable cardinals and which can be established within the theory ZFC.
Remark 2. We see, in particular, that if the Continuum Hypothesis holds, then the cardinality of the continuum is not real-valued measurable, either. It is reasonable to recall here that the latter result was first obtained by Banach and Kuratowski in their old joint paper [i'].Actually, the method of [7] gives a more general result. Namely, let us consider the family F of all functions acting from w into w . Let f and g be any two functions from F . We put f 5 g if and only if there exists a natural number n = n(f,g) such that f(m) 5 g(m) for all natural numbers m 2 n. Obviously, the relation 5 is a preordering of F. Now, if the Continuum Hypothesis holds, it is not difficult to define a subset of F satisfying the following two conditions: a) if f is an arbitrary function from F, then there exists an ordinal 4 < w l such that f 3 ft; b) for any ordinals ( and such that E < 6 < w l , the relation fC3 ft is not true. Evidently, each of conditions a) and b) implies the equality
<
Further, for any two natural numbers m and n, we put
So we get a double family of sets
which is usually called the Banach-Kuratowski matrix. It is easy to check that, for each m < w , we have the inclusions
and the equality
E = U{E,,,
: n
< w).
Also, conditions a) and b) immediately imply that if f is an arbitrary function from F ,then the intersection
is at most countable. From these properties of the Banach-Kuratowski matrix it is not hard to deduce that there does not exist a nonzero a-finite diffused measure on E defined simultaneously for all sets
In addition, it can easily be seen that an analogous result is true for many other functionals (much more general than measures). Namely, let v be a real-valued positive (Lea nonnegative) function defined on some class of subsets of El closed under finite intersections. We say (cf, the corresponding definition presented in Chapter 9) that v is an admissible functional (on E) if the following conditions hold: 1) the family of all countable subsets of E is contained in dom(v) and, for any countable set Z c E , we have the equality v(Z) = 0; 2) if {X, : n < w ) is an increasing (with respect to inclusion) family of sets, such that Xn E dom(v) for all n < w , then the set U{X, : n < w) also belongs to dom(v), and
3) if {Yn : n < w ) is a decreasing (with respect to inclusion) family of sets, such that Yn E dom(v) for all n < w , then the set n{Y, : n < w ) also belongs to dom(v), and v(n{Y, : n
< w ) ) 1 in f{v(Y,)
: n
< w).
Evidently, if v is a finite diffused measure on E , then v satisfies conditions I), 2) and 3). But, in general, an admissible functional v need not have any additive properties similar to the corresponding properties of usual measures. However, the Banach-Kuratowski method works for such
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functionals, too, and we can conclude that there does not exist a nonzero admissible functional on E defined simultaneously for all sets of the BanachKuratowski matrix.
Exercise 4. Let E be a set with card(E) Kuratowski matrix (Em,n)na<w,n<w
> w for which a Banach-
of subsets of E can be constructed. Prove that there does not exist a nonzero admissible functional v on E satisfying the relation (Vm < w)(Vn < w)(E,,,
E dom(v)).
Let us return to the Sierpiriski partition and consider some other interesting facts related to it. Assuming the Continuum Hypothesis, there exists a function
such that
< w) ( n < w) are some motions of the
R~ = U{gn(I'(q5)) : n
where r(Q) denotes the graph of Q and g, plane R 2 , each of which is either a translation or a rotation (about a point) whose angle is equal to h / 2 . The proof of this result is not difficult and we leave it to the reader.
Exercise 5. By starting with the Sierpiliski partition of R ~ give , a detailed proof of the above result. Let now X and Y be any two sets. We recall (see, e.g., Chapter 0) that a set-valued mapping is an arbitrary function of the type
where P(Y) denotes, as usual, the family of all subsets of Y. According to a well-known definition from general set theory, a subset Z of X is independent with respect to F if, for any two distinct elements x E Z and y E 2,we have the relations
It can easily be shown that there exists a set-valued mapping
such that no two-element subset of wl is independent with respect to F . In fact, the desired set-valued mapping F may be defined as follows:
where A is the first set of the Sierpiriski partition {A, B) of w l x w l . In connection with this fact, let us remark that if a set-valued mapping
is given, then there always exists a subset 3 of wl satisfying the following two conditions: 1) card(=) = w l ; 2) E is independent with respect to F . The reader can easily derive this result from the so-called A-system lemma (see, e.g., [MI). It is also reasonable to point out here that the A-system lemma is a theorem of the theory ZF & DC. Finally, let us mention that an analogous result (concerning the existence of large independent subsets) holds true for uncountable cardinal numbers, but the proof of this generalized result due to Hajnal and Erdos is more difficult and needs an additional argument. There are many other interesting statements and facts which are related to the Sierpiriski partition or can be obtained by using certain properties of this partition (see, e.g., works [27], [28], [46], [73], [84], [108], [109], [134], [I351 and [144]). Here we wish to discuss only one application of the Sierpiriski partition in real analysis. Namely, we shall present one result of Morayne [I081 which establishes an interesting connection of this partition with the existence of some strange mappings acting from R onto R x R. In order to present the above-mentioned result, we need several auxiliary notions and facts. Let f be a partial function acting from R into R, and let X be a subset of R. We say that f satisfies the Banach condition on X if the set
{Y
E f ( X ) : card( f-'(y) f~ X)
is of Lebesgue measure zero.
> w)
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10
We also recall that a partial function f acting from R into R satisfies the Lipschitz condition if there exists a real L 2 0 such that
for all u and v belonging to dom(f). In this case, the constant L is usually called a Lipschite constant for f . It is not hard to prove the following auxiliary statement.
L e m m a 1. Let (X, d) be a metric space, let Y be a subset of X and let f be a function acting from Y into R and satisfying the Lipschitz condition with a Lipschitz consiant Lip(f) = L 2 0. Then f can be extended to a function
g : X+R which fulfils thas condition, too, with the same Lipschitz constant L. Proof. Assume that Y
# 0 and, for any point x E X , define
g(x) = in f {f (y)
+ Ld(x, Y) : y E Y).
In this way we obtain a mapping g from X into R . Let us check that g is the required extension o f f . Fix an arbitrary x E Y. Obviously, for each y E Y, we have g(x) 5 f (?4) Ld(8J ?4).
+
In particular, putting y = x, we get
On the other hand, the relation
implies that
f (x) 5 f (?4)+ Ld(x, ?4)
(?4 E Y)
and, hence, f ( x ) 5 g(x). Consequently, g(x) = f (x) and g is an extension of f . Now, let X I and xz be two arbitrary points from X and let exist points yl E Y and yz E Y such that
E
> 0. There
Then we may write
and, finally, 19(xi) - 9(~2)15 L d ( z l 9 ~ 2 ) Since c is an arbitrary strictly positive number, we conclude that
which finishes the proof of Lemma 1. Actually, we need only a very special case of this lemma when X = R. In this case, the proof can be done directly.
.
Lemma 2. Let f be a partial function acting from R into R, and suppose that f is differentiable at all points of dom(f), i.e. for each point t E dorn(f), there exists a derivative fi,,(f)(t) relative t o the set dom(f). Then the domain o f f can be represented an the form
where card(I) _< w and all sets Pi have the property that f lPi satisfies the Lipschitz condition.
Proof. First, let us denote
and, for any natural number n
> 0, define the set
D, = {t E D : (Vt' E D)(ltt - t (
5 l/n
D, by
=+ (f(t') - f (t) 1 < nlt' - to}.
Then, by taking account of the assumption that f is differentiable relative to D , it is not hard to check the equality
Further, for each natural n
> 0, we may write
CHAPTER 10
where
(Vk
< w ) ( d i a m ( D n k )5 l / n ) .
Now, it immediately follows from the definition of D , that all the restrictions f)Dnk (O
and Lemma 2 is thus proved. The next auxiliary proposition is well known in real analysis and is due to Banach (see, e.g., [ I l l ]or [127]).
Lemma 3. Let f be a continuous real-valued function offinite (bounded) variation, defined on some segment [a, b] C R. For each y E R, we put
af c a r d ( f - ' ( y ) ) 2 w , and
if card( f - l ( y ) )
< w . Then the function
is integrable in the Lebesgue sense, and the relation
is valid. In particular, for almost all (with respect t o the Lebesgue measure) points y E R, we have card( f - ' ( y ) ) < w. Exercise 6, Give a proof of Lemma 3. Lemma 4. Let f be a functaon actang from R into R, and let D = {t E R : ft(t) exists). Then f satisfies the Banach condition on D .
Proof. In view of Lemmas 1 and 2, it suffices to demonstrate that, for each closed bounded interval [a, b] C R, any function
satisfying the Lipschitz condition on [a, b], fulfils the Banach condition on the same interval. But this immediately follows from Lemma 3 because f * is continuous and of finite variation on [a, b]. Now, we are ready to formulate and prove the result of Morayne [108]. Actually, this result yields a purely analytic equivalent of the Continuum Hypothesis.
Theorem 1. The following two assertions are equivalent: 1) the Continuum Hypothesis; 2) there exists a surjection
such that, for any point t E R, at least one of the coordinate functions fl and fi is diferentiable at t .
Proof. We first establish implication 1 ) 2). Suppose that the Continuum Hypothesis holds. Then we may consider a Sierpiliski type partition { A ,B ) of R x R such that
<
(Vx E R ) ( c a r d ( A ( z ) ) w ) ,
(Vy E R ) ( c a r d ( B ( y ) )5 w )
where, as usual,
Further, we introduce a function
defined by the formula 4 ( t ) = t sin(t) a
(t E R).
It is evident, from the geometrical point of view, that, for any two numbers u E R and v E R, the sets
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are countably infinite. So we may write 4 - l ( ~ )n ] - W , -11 = {ty,t;,
...,t;,
...),
4 - ' ( ~ ) n [ 1 , + m [ = {s;,s;,~..,s;,...). At the same time, we can represent the countable sets A(u) = {y E R : (u, y) E A), B ( v ) = { x E R : (x,v) E B ) in the following form:
B(v) = {by,bil
.#.,
b;, ...},
Let now t be an arbitrary point of R . If t E ] - oo, 1[, then we put
If t E [I, +oo[, then t = s i for some real v and natural n. In this case, we define f l ( t ) = fl(s;) = b ; . Analogously, if t E ] - 1, +m[, then we put
If t E ] - oo, -11, then t = t; for some real u and natural n. In this case, we define f2(t) = f2(tun) = a;. Finally, we introduce a mapping
by f = (fi, f2). Let us verify that f is the required function. For this purpose, take any point t E R. Since
there exists a neighbourhood W(t) of t such that
and, hence, a t least one of the coordinate functions f l and f 2 coincides with the function 4 on W ( t ) . But 4 is differentiable everywhere on R. Consequently, at least one of the functions f l and fi is differentiable on W ( t )and, in particular, differentiable a t the point t. Also, it can easily be checked that f is a surjection. Indeed, let ( u , v) be an arbitrary point of R ~Since . the equality
holds, the point (u, v) belongs either to A or to B . We may assume, without loss of generality, that (u, v) E A. Then
~-I(U)
n ] - CO, -11
= it;:,
ti,...,t;,
...)
and, for some natural number n, we must have
Putting t = tU, and taking into account the definition of f , we get
which shows that our f is a surjection. In this way we have established implication 1) =+ 2 ) . Suppose now that assertion 2) is valid, i.e. for some surjective mapping
the equality
R = Dl U Dz is fulfilled, where
Dl = {t E R : f i ( t ) exists}, D2 = {t E R
:
f i ( t ) exists).
In conformity with Lemma 4, the coordinate functions f l and f2 satisfy the Banach condition on the sets D l and D 2 , respectively. Hence the sets
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are of Lebesgue measure zero. Let us put
Then, for M1 and Mz, we have the equalities
Now, we define
A = f ( ~ n~(MI ) x MZ),
= fp2> n (MI x MZ). Taking account of the fact that f is a surjection, we infer
Also, it immediately follows from the definition of the sets M1 and Mz that
(VY E M~)(card(B(Y))5 w)In other words, we obtain a covering { A , B ) of the product set M1 x M2, having properties very similar to those of the Sierpiliski partition of wl x w l . Since the product set M1 x M2 may be identified (only in the purely settheoretical sense) with R x R, we conclude that the Continuum Hypothesis must be true (cf. the argument presented at the beginning of this chapter, or Exercise 1). This ends the proof of Theorem 1. R e m a r k 3. One interesting generalization of the above theorem was obtained in paper [27]. The function f considered in Theorem 1 is singular from the point of view of Lebesgue measurability. The following exercise illustrates this fact. Exercise 7. Let f = ( f l , f i ) be any surjection from R onto R x R having the property that, for each t E R, at least one of the coordinate functions f l and f 2 is differentiable at t . Show that f is not measurable in the Lebesgue sense. In connection with this exercise, let us note that a stronger result is contained in [108]. Exercise 8. Let n 2 2 be a natural number. We consider the ndimensional' Euclidean space R n consisting of all n-sequences x of the form x=(x1,x2,...,xn)
( x ~ E R i,= 1 , 2 ,...,n).
S I E R P I ~ S K I ' SPARTITION OF
THE EUCLIDEAN PLANE
For any t E R and for each natural index i E [I, n], we denote
Obviously, r i ( t ) is a coordinate hyperplane in the space R n . Demonstrate that the following two assertions are equivalent: 1) the Continuum Hypothesis; 2) there exists a partition {A1,Az,..., A,) of R n such that, for any t E R and for each natural i E [ l , n ] , the inequality
is satisfied. This result generalizes the case n = 2 considered above and is also due to Sierpiliski. The following simple exercise shows that, for the infinite-dimensional analogues of Euclidean space, the situation is essentially different. Exercise 9. Let us consider the infinite-dimensional topological vector space RW = R1 x Rz x ... x Rn x ... where (Vi E w
\ {O))(Rj = R ) ,
and, as above, denote by r d ( t ) the coordinate hyperplane in this space, corresponding to an index i E {1,2, ..., n , ...)and to a real t E R. Show that there does not exist a covering
of RW such that, for any i E w
\ (0)
and for each t E R, the relation
is fulfilled. Consequently, there is no covering {Ba : i E w \ (0)) of RW such that, for any i E w
is satisfied.
\ (0)
and for each t E R, the relation
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The results presented in the next two exercises are due to Morayne (see [I081 and [log]). Exercise 10. Let n 2 2 be a natural number. Prove that the following two assertions are equivalent: 1) the Continuum Hypothesis; 2) there exists a surjective mapping
such that, for any point t E R, a t least one of the coordinate functions f l , f 2 , ... , f n is differentiable at t . Exercise 11. Show that there is no surjection
having the property that, for each point t E R, at least one of the coordinate functions f l , f 2 , ... , f n , ... is differentiable at t . We want to finish this chapter with one statement closely related to the Sierpiriski partition of wl x w l . This statement does not require additional set-theoretical hypotheses and establishes a certain relationship between Sierpiliski type partitions and measurability in the Lebesgue sense. T h e o r e m 2. In the theory ZF & DC, the assertion
"there exists a well - ordering of R" inaplies the asseriaon "there exists a subset of R nonmeasurable i n the Lebesgue sense".
Proof. Obviously, the existence of a well-ordering of R means that R can be represented as an injective family
where a denotes some ordinal number of cardinality continuum. Also, it is clear that, in order to prove the existence of Lebesgue nonmeasurable subsets of R, it suffices to establish the existence of subsets of the plane R 2 , nonmeasurable with respect to the standard two-dimensional Lebesgue measure on R 2 . Let Xz=XxX
denote the latter measure (where A is the standard Lebesgue measure on R ) . Let p _< a be the least ordinal for which
where A * is the outer measure associated with A . If the set {xE : 4 < p) is nonmeasurable with respect to./\,then we are done. Otherwise, we may write {XE : E < PI E dom(A), A({x, : 4 < PI) > 0 and, according to the definition of PI for each ordinal 7 < P, we have
Consider now a subset Z of R 2 defined as follows:
We assert that Z is nonmeasurable with respect to A 2 , Indeed, suppose for a moment that Z E dom(A2). Then, considering the vertical and horizontal sections of Z and applying the classical Fubini theorem to Z , we get, on the one hand, the relation Az(Z) > 0 and, on the other hand, the equality
Since this is impossible, we conclude that Z is not X2-measurable, which also implies the existence of a Lebesgue nonmeasurable subset of the real line. Exercise 12. By applying a certain topological analogue of the classical Fubini theorem - the so-called Kuratowski-Ulam theorem (see, e.g., [89], [I101 or [117]) - prove an analogue of Theorem 2 for the Baire property. More precisely, demonstrate in the theory Z F & DC that the assertion "there exists a well - ordering of R " implies the assertion "there exists a subset of R without the Baire property".
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10
In other words, Theorem 2 and the result of the preceding exercise show us that the existence of a well-ordering of the real line R immediately yields the existence of subsets of R having very bad descriptive structure (from the points of view of Lebesgue measurability and the Baire property). In this connection, let us indicate that the existence of a totally imperfect subset of R of cardinality continuum also implies (in the same theory ZF & DC) the existence of a Lebesgue nonmeasurable subset of R and the existence of a subset of R without the Baire property.
11. Sup-measurable and weakly sup-measurable functions
It is well known that the notion of measurability of sets and functions plays an important role in various fields of classical and modern analysis (also, in probability theory and general topology). For functions of several variables, a related notion of sup-measurability was introduced and investigated (see, e.g., [133], [85], [52], [77], [3] and the references given therein). It turned out that this notion can successfully be applied to some topics from analysis and, in particular, to the theory of ordinary differential equations (for more information concerning applications of sup-measurable mappings in the above-mentioned theory, see [85] and the next chapter of this book). Here we shall introduce and examine the following three classes of functions acting from R x R into R: 1) the class of sup-continuous mappings; 2) the class of sup-measurable mappings; 3) the class of weakly sup-measurable mappings. We begin with sup-continuous and sup-measurable functions. We shall say that a mapping
is sup-continuous (sup-measurable) with respect to the second variable y if, for every continuous (Lebesgue measurable) function
the superposition (Z,# :
R+R
given by the formula
is also continuous (Lebesgue measurable).
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Let us mention that, actually, the first notion yields nothing new: it turns out that the class of all sup-continuous mappings coincides with the class of all continuous mappings acting from R x R into R. For the sake of completeness, we present here the proof of this simple (and probably well-known) fact.
Theorem 1. Let (Z, be a mapping acting from R x R into R. Then the following two assertions are equivalent: 1) (Z, is continuous; 2) (Z, is sup-continuous. Proof. The implication 1) =$ 2) is trivial. So it remains to establish only the converse implication 2) =+ 1). Let (Z, be sup-continuous, and suppose that (Z, is not continuous. Then there exist a point (xo,yo) of R x R, a real number c > 0 and a sequence of points
such that: a) lamn-+co (xn ~ n =) (XO, YO ); b) 1@(xnI~ n-) @(xo,yo)l > E for all n E N \ (0). We may assume, without loss of generality, that the sequence of points
is injective and x,
# xo for each n E N \ (0). Indeed, if
denotes the function identically equal to yn, then the function
is continuous and (Z,j,(x,) = (Z,(x,, y,). Therefore, for some strictly positive real number 6 = 6(xn) and for all points x belonging to the open interval ]xn - 6, x, + 6[, we have the inequality
or, equivalently, I@(x,yn) - @(xo,YO)/> c From this fact it immediately follows that the above-mentioned sequence {x, : n E N , n > 0) can be chosen injective and satisfying the relation
S U P - ME A S U R A B L E AN D W E A KLY S U P - ME A S U R A B L E F UN C TI O N S
219
Now, it is not difficult to define a continuous function f : R+R such that
(Vn E N)(f(xn) = ~ n ) . For this function f , we get the continuous superposition
Since limn,+,
2,
= XO,
we must have the equality
and, consequently,
which is impossible. This contradiction finishes the proof of Theorem 1.
A completely different situation exists for sup-measurable mappings. On the one hand, simple examples show that if
is a Lebesgue measurable mapping, then it need not be sup-measurable.
Exercise 1. Give an example of a function
which is Lebesgue measurable but is not sup-measurable. More precisely, demonstrate that the existence of such examples follows directly from the widely known fact that the composition of Lebesgue measurable functions (acting from R into R) need not be Lebesgue measurable. On the other hand, it turns out that there exist (under some additional set-theoretical axioms) various sup-measurable mappings which are not measurable in the Lebesgue sense. In order to present this result, let us first formulate and prove one simple auxiliary statement.
Lemma 1. Suppose that 9 is a mapping acting from R x R anto R. Then the following two assertions are equivalent:
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CHAPTER
is sup-measurable; 2) for every contanuous function $ : R Lebesgue measurable.
11
1)
-+
R, the function q3, is
Proof. The implication 1) j 2) is trivial. Let us show that the converse implication 2) j 1) is true, too. Let 9 satisfy 2) and let $ be an arbitrary Lebesgue measurable function acting from R into R. Applying the theorem of Luzin, we can find a countable partition {Xk : k < w) of R and a countable family {$k : k < w ) of functions from R into R , such that: a) all sets Xk (1 5 k < w) are closed in R and Xo is of Lebesgue measure zero; b) all functions $k (1 5 k < w ) are continuous; c) for each index k < w , the restriction of $ to Xk coincides with the restriction of $k to X k . Let us denote by x k the characteristic function of X k . Then it is not difficult to check the equality
According to our assumption, all superpositions 9@k (1 5 k < w) are Lebesgue measurable. In addition, the function xo.9@,is equivalent to zero. Thus, we easily conclude that the superposition 91C, is Lebesgue measurable, too. Exercise 2. Let (Z, be a mapping acting from R x R into R. Suppose that this mapping satisfies the so-called Carathdodory conditions, i.e., 1) for each x E R, the partial function
is continuous; 2) for each y E R , the partial function
is Lebesgue measurable. Show that: a) (Z, is measurable with respect to the usual two-dimensional Lebesgue measure on the plane; b) (Z, is sup-measurable. Exercise 3. Let (Z, be a mapping acting from R x R into R . Suppose that the following two conditions hold:
SUP- MEASURABLE A N D WEAKLY SUP- MEASURABLE FUNCTIONS
22 1
1) for each x E R , the partial function
is a continuous mapping from R into R ; 2) for each y E R, the partial function
is a Borel mapping from R into R. Show that (Z, is a Borel mapping from R x R into R . By using the method of transfinite induction, give an example of a function 9 acting from R x R into R such that: a) for any points xo E R and yo E R , the partial functions
X
+
* ( x , yo)
(x E R)
are Borel (more precisely, are of first Baire class); b) 9 is not measurable with respect to the standard two-dimensional Lebesgue measure on the plane. Exercise 4. Let n > 1 be a natural number and let (Z, be a mapping acting from the n-dimensional Euclidean space R n into R . Suppose also that @ satisfies the following condition: for each natural index i E [I,n] and for any points
the partial function
is continuous. Prove, using induction on n, that @ is a Borel mapping from Rn into R (more precisely, prove that the Baire class of (Z, is less than or equal to n 1).
-
The next exercise shows that, for some standard infinite-dimensional spaces, the situation is essentially different from the one described in the previous exercise. Exercise 5. Let T denote the one-dimensional unit torus, i.e, the set T = { ( x , y ) E R x R : x2
+ y 2 = 1)
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11
is regarded as a commutative compact topological group with respect to the usual group operation and the Euclidean topology. We denote by e the neutral element of this group. Now, consider the product group T W . It is a commutative compact topological group, too. Equip this product group with the Haar probability measure p . In fact, p is the product measure of a countable family of measures, each of which coincides with the Haar probability measure on T. Further, denote by G the subset of T W consisting of all those elements { x , : n E w ) E T W for which we have
card({n E w : x ,
# e ) ) < w.
Obviously, G is a dense Bore1 subgroup of T W . Finally, let {Di : i E I) be the injective family of all G-orbits in T W . Check that
where c denotes, as usual, the cardinality of the continuum. Prove that there exists a subset J of I such that the set
is not measurable with respect to the completion of the Haar measure p . Deduce from this fact that there exists a mapping
satisfying the relations: a) r a n ( 9 ) = ( 0 , l ) ; b) 8 is constant with respect to each variable x , ( n E w); in particular, 9 is continuous with respect to each x,; c) 8 is nonmeasurable with respect to the completion of p .
Exercise 6. Show that the result presented in the previous exercise has a direct analogue in terms of the Baire property. Now, we wish to formulate and prove the following statement.
Theorem 2. Let X denote the standard Lebesgue measure on R and let [RIcC be the family of all subsets of R, whose cardinalities are strictly less than c. There exists a subset Z of R x R such that: 1) no three distinct points of Z belong to a straaght line (in other words, Z is a set of points in general position); 2) Z is a Lebesgue nonmeasurable subset of R x R; 8) af [R]<' c dom(X), then the characteristic jknction of Z is supmeasurable.
SUP- MEASURABLE A N D WEAKLY SUP- MEASURABLE
FUNCTIONS
223
Proof. The argument is very similar to the one applied in the construction of a Sierpiliski-Zygmund function (see Chapter 5). Obviously, we can identify c with the first ordinal number a such that card(a) = c. Let X2 denote the standard two-dimensional Lebesgue measure on the plane R x R and let {ZE : < a ) be the family of all Bore1 subsets of R x R having a strictly positive Xz-measure. In addition, let {4( : E < a) be the family of all continuous functions acting from R into R.As usual, we identify any function from R into R with its graph lying in the plane Rx R.Now, using the method of transfinite recursion, we are going to define an a-sequence of points { ( ~ t , Y t ): E < a ) c Rx R, satisfying the following conditions: a) i f < < a , C < a a n d < # C , then xt # xc; b) for each ( < a , the point (x5, yt) belongs to the set ZI; c) for each < a, the point (xt, ye) does not belong to the union of the family {4( : C I El; d) for each < a, no three distinct points of the set { ( x ~ye) , : C 5 <) belong to a straight line. Suppose that, for an ordinal < a , the partial <-sequence of points {(xC,yc) : C < 4) has already been defined. Let us consider the set Zt. We have Xz(Zt) > 0.
<
< <
<
According to the classical Fubini theorem, we can write
where ZE(x) denotes the section of Zt corresponding to a point x E R. Taking account of the latter formula, we see that there exists an element
for which X(Zt(xf)) > 0. In particular, we get the equality
Consequently, there exists an element
Moreover, ye can be chosen in such a way that the corresponding point (xe, ye) does not belong to the union of all straight lines having at least two common points with the set {(xe, yC) : C < 4).
C H APTE R 11
224
We have thus defined the point (x,, ye) E R x R . Proceeding in this manner, we are able to construct the a-sequence {(x,, y E ) : 4 < a ) satisfying conditions a), b), c) and d). Finally, let us put and let @ denote the characteristic function of Z (obviously, Z is considered as a subset of the plane R x R ) . Notice that Z can also be regarded as the graph of a partial function acting from R into R . Hence the inner Azmeasure of Z is equal to zero. On the other hand, the construction of Z immediately yields that Z is a Xz-thick subset of the plane. Consequently, Z is nonmeasurable in the Lebesgue sense and the same is true for its characteristic function @. It remains to check that @ is a sup-measurable mapping under the assumption
Let us take an arbitrary continuous function for some ordinal 4 < a. Now, we can write
4
: R -+ R. Then
4 = dC
and it easily follows from condition c) that card({x E R : (x, 4((x)) E 2 ) ) _< card([)
+w < c.
Since the inclusion [R]
S U P - M E A S U R A B L E A N D W E A KLY S U P - M E A S U R A B LE F U N C TI O N S
satisfying the following conditions: 1) for every Lebesgue measurable function $ : R tions x -,@($(x),2) (a: E R ) ,
-+
225
R , the superposi-
are also Lebesgue measurable; 2) O is not measurable in the Lebesgue sense, R e m a r k 3. In fact, for the existence of a function O of the previous exercise, we do not need the whole power of Martin's Axiom. It suffices to apply a certain set-theoretical hypothesis weaker than Martin's Axiom (cf. Theorem 3 below). On the other hand, it is not difficult to prove that if a mapping
has the property that, for any two Bore1 functions f and g acting from R into R , the superposition
is Lebesgue measurable, then iP is Lebesgue measurable, too (see, e.g., [77]). The next exercise contains a slightly more general result.
-
Exercise 8. Let \E be a function acting from R x R into R, such that the superposition 3 q(f(.),g(x)) (3 E R ) is Lebesgue measurable for all continuous functions f and g acting from R into R. Show that @ is Lebesgue measurable (cf. Lemma 1). We now introduce the notion of a weakly sup-measurable function. Let O be a mapping acting from R x R into R . We shall say that O is weakly sup-measurable if, for any continuous function $ : R -t R differentiable almost everywhere (with respect to A), the superposition O+ is Lebesgue measurable. Evidently, from the point of view of the theory of ordinary differential equations, the notion of a weakly sup-measurable mapping is more important than the notion of a sup-measurable mapping, because any solution of an ordinary differential equation must be continuous and differentiable almost every where. Clearly, Theorem 2 can be formulated in terms of weakly sup-measurable mappings. In this connection, the following question arises naturally: does
226
C H A PT E R
11
there exist a weakly sup-measurable mapping which is not sup-measurable? In order to give a partial answer to this question, we need one auxiliary statement due to ~ a r n i k(see [60]). Lemma 2. There exists a continuous function f : R nowhere approximately differentiable.
-+
R which i s
We recall that Lemma 2 was proved in the chapter of our book devoted to nowhere approximately differentiable functions (see Chapter 4). We also want to recall that, in fact, ~ a r n i kproved in [60] that the set of all those functions from the Banach space C([O, I]), which are nowhere approximately differentiable, is residual in C([O, I]), i.e. is the complement of a first category set. Nevertheless, in our further considerations, we need only one such function. Theorem 3. Suppose that: 1) [R]<' C dom(X); 2) for any cardinal number IC < c and for any family {XE : A-measure zero subseis of R , we have
E < K)
of
Then there exists a weakly sup-measurable mapping O whach is not supmeasurable.
Proof. We can identify c with the first ordinal number a such that card(a) = c. Let f be a function from Lemma 2. Let {BE : 5 < a} be some Bore1 base of the o-ideal of all Lebesgue measure zero subsets of R and let {q+ : 5 < a} be the family of all continuous functions acting from R into R and differentiable almost everywhere in R . We are going to construct (by transfinite recursion) an injective a-sequence
of points belonging to the graph of f . Suppose that, for an ordinal 5 < a, the partial &sequence {(xC,yS) : C < [) has already been defined. Notice that, for each C 5, the closed set
<
is of Lebesgue measure zero. Indeed, assuming otherwise, i.e. X(PC) > 0, we can find a density point x of PCbelonging to PCsuch that there exists an approximate derivative f&(x). But this is impossible. Consequently, X(P0 = 0 for all C 4, and the set
<
S U P - M E A S U R A B L E A N D W E A KLY S U P - ME A S U R A B L E F U N C T I O N S
227
is not empty. Let x5 be an arbitrary point from this set and let yt = f ("5). Proceeding in such a manner, we are able to define the required family of points {(xe,y5) : 4 < a). Now, we put
and denote by O the characteristic function of Z. Then it can easily be seen that O is a weakly sup-measurable mapping (cf, the proof of Theorem 2). On the other hand, let us consider the superposition O f . Obviously, we have O j ( t ) = 1*(x, f(x)) E Z * x € X . It follows from our construction that X is a Sierpiriski type subset of the real line R (for the definition and various properties of Sierpiriski sets, see e.g. [89],[107], [110], [I171 or that chapter of our book which is specially devoted to Luzin and Sierpiriski sets). In particular, as shown in that chapter, X is not measurable in the Lebesgue sense and, therefore, Of is not Lebesgue measurable, either. We thus conclude that O is not a supmeasurable mapping.
Remark 4. It is well known that assumptions 1) and 2) of Theorem 3 are logically independent (see, for instance, [87]). Slightly changing the argument presented above, one can show (under the assumptions of Theorem 3) that there exists a weakly sup-measurable mapping which is not sup-measurable and, in addition, is not Lebesgue measurable. We do not know whether the assertion of Theorem 3 is valid in the theory ZFC.
Remark 5. Evidently, the notion of sup-measurability can be formulated in terms of the Baire property instead of measurability in the Lebesgue sense. More precisely, we say that a function
is sup-measurable in the sense of the Baire property if, for any function
having the Baire property, the superposition
has the Baire property, too. It is not difficult to verify that, for the Baire property, a direct analogue of Theorem 2 holds true. The corresponding analogue of Theorem 3 also
228
C HAPTER 1 1
holds (in this case, we do not need Lemma 2; it suffices to apply the existence of a continuous nowhere differentiable function acting from R into R ) . The corresponding details are left to the reader as a useful exercise.
Exercise 9. Let k be a strictly positive natural number. Suppose that: a) any subset X of R with card(X) < c has the Baire property, i.e. is of first category; b) for any family { X i : i E I) with card(1) < c, consisting of first category subsets of R , we have R
# U{Xi
: i E I).
Show that there exists a mapping
satisfying the following relations: 1) for every k-times continuously differentiable function
the superposition 3 @(x,g(x)) (8 E R ) has the Baire property; 2) there is a (k - 1)-times continuously differentiable function +
such that the superposition
does not have the Baire property. Formulate and prove an analogous result in terms of the Lebesgue measure.
Exercise 10. Let @ be a function acting from R x R into R . Show that the following two assertions are equivalent: a) O is sup-measurable in the sense of the Baire property; b) for every Bore1 function 4 acting from R into R , the superposition
has the Baire property.
S U P - ME A S U R A BLE A N D W E A KLY S U P - ME A S U R A BLE F U N C TI O N S
Also, determine the precise Baire order of equivalence of these two assertions.
229
4 which is sufficient for the
Exercise 11. Prove an analogue of Theorem 2 for the Baire property. Exercise 12. Prove an analogue of Theorem 3 for the Baire property. As mentioned above (see Exercise 2 of this chapter), the functions of two variables, satisfying the Carathkodory conditions, are good from the point of view of sup-measurability. In addition, such functions play an important role in the theory of ordinary differential equations. The following definition introduces a slightly more general class of functions. Let (X, S,p) be a space with a complete probability measure, let Y be a topological space and let
be a function. We say that f almost satisfies the Carathkodory conditions if 1) for almost all (with respect to p ) points x E X , the partial function f (x, -) is continuous on Y ; 2) for all points y E Y, the partial function f (., y) is p-measurable. Obviously, if f satisfies the Carathdodory conditions, then it almost satisfies these conditions. The converse assertion is not true, in general. However, it can easily be verified that iff almost satisfies the Carathdodory conditions, then there exists a function
satisfying these conditions, such that
for almost all points x E X . Some useful additional information about functions which almost satisfy the Carathkodory conditions is given in the exercises below. Exercise 13. Let ( X , S , p ) be a space with a complete probability measure, let Y be a topological space with a countable base and let
be a function satisfying the Carathkodory conditions. Pick a countable base {U, : n E N ) of Y and consider the family F of all those functions 4 which can be represented in the form
CHAPTER
11
where n E N , qn€Qn[O,l] and XU,, is the characteristic function of the set Un. Since the family F is countable, we may write
Further, for any k E N , let us define
Fix a countable subset Yo of Y dense in Y and, for each k E N , put
Check that the set Xi is p-measurable and the equality Xk = Xi holds. Therefore, Xk is p-measurable. Show that f ( 3 , ~= ) sUpkENX X , ( X ) $ ~ ( Y ) for all x E X and y E Y. Deduce from this fact that: a) the function f is measurable with respect to the product of a-algebras S and B(Y), where B(Y) denotes, as usual, the Bore1 a-algebra of Y; b) the function f is sup-measurable, i.e, for any p-measurable mapping
the superposition x f (3,$(XI) ( 3 E X ) is p-measurable, too. Suppose, in addition, that X is a Hausdorff topological space and p is the completion of a Radon probability measure on X . By applying a Luzin type theorem on the structure of p-measurable real-valued functions, prove that, for each E > 0, there exists a compact set P C X with p ( P ) > 1 - E , such that the restriction of f to the product set P x Y is lower semicontinuous. Extend the results presented above to the functions which almost satisfy the Carathkodory conditions. +
Exercise 14. Let X be a Hausdorff topological space, let p be the completion of a Radon probability measure on X and let Y be a topological space with a countable base. By applying the results of Exercise 13, show that, for a function f : X x Y -+ [O,11,
SUP-MEASURABLE A N D WEAKLY SUP- MEASURABLE
FUNCTIONS
23 1
the following two assertions are equivalent: a) f almost satisfies the Carath6odory conditions; b) for any E > 0, there exists a compact set P E X with p(P) > 1 - E , such that the restriction o f f to the product set P x Y is continuous. Note that the equivalence of these two assertions is usually called the Scorza Dragoni theorem.
Exercise 15. Let (X, S , p ) be a space with a complete probability measure, let Y be a locally compact topological space with a countable base and let Z be a subset of the product space X x Y, measurable with respect to the product of a-algebras S and B(Y). Demonstrate that the set pri(Z) = {a: E X : (3y E Y)((x,u) E 2 ) ) is measurable with respect to p.
Exercise 16, Let ( X , S , p ) be a space with a complete probability measure, let Y be a nonempty compact metric space and let
be a function satisfying the Carathkodory conditions. For each point x E X , denote F(x) = {y E Y : f ( x , y ) = i n f t ~ y f ( 3 , t ) ) . Check that F(x) is a nonempty closed subset of Y. Hence we have a setvalued mapping F : X + P(Y). Prove that, for any open set U
E Y, the set
is p-measurable. Derive from this fact, by using the theorem of Kuratowski and Ryll-Nardzewski on the existence of measurable selectors (see [go] or [92]), that there exists a p-measurable mapping
such that f ( x , h(x)) = infyey f ( x , !I) for all points x E X . Formulate and prove an analogous statement for functions almost satisfying the Carathkodory conditions.
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CHAPTER 11
Let us present a result which is much deeper than that given in the exercise above. For this purpose, we need the notion of a C-set. This notion was introduced (by Kolmogorov) many years ago and was thoroughly investigated by several authors (see, e.g., [128]). Let E be a metric space. We define the family of all C-sets in E as the smallest class containing all open subsets of E and closed under the operation of taking the complement and under the (A)-operation. Since the (A)-operation includes in itself countable unions and countable intersections, we see that the class of all C-sets in E forms a certain a-algebra containing all analytic subsets of E (in particular, the Borel a-algebra of E is contained in the class of all C-subsets of E). It immediately follows from the definition that all C-sets are universally measurable and possess the Baire property in the restricted sense. Let now El be another metric space and let
be a mapping. We say that f is C-measurable if, for each open subset U of El, the preimage f 'l (U) is a C-set in E. From the definition of the (A)-operation follows that the preimage of the result of this operation over a given family of sets coincides with the result of the same operation over the family of preimages of sets. Taking this fact into account, we may easily infer that, for any C-measurable mapping f from E into El and for any C-set Z in E', the preimage f q l ( Z ) is a C-set in E. In particular, the composition of two C-measurable functions is a Cmeasurable function, too (this is an important feature of C-measurable functions which has no analogue e.g. in the class of Lebesgue measurable functions). Let now X and Y be any two Polish spaces and let
f : X x Y -+[O,1] be a Borel mapping such that (Vx E X)(3y E Y)(f(x,y) = inf,,yf(x,t)). Let us define . f * ( x ) = i n f t ~ ~ f ( x , t )( x E X ) .
Note that, for each a E R, we have
S U P - M E A S U R A B L E A N D W E A KL Y S U P - ME A S U R A BL E F UN C TI O N S
233
Since the original function f is Borel, we conclude that the first set in the equality above is analytic, i.e, the function f * is measurable with respect to the a-algebra generated by all analytic subsets of X (hence f * is Cmeasurable as well). In the product space X x Y x R consider the set
Obviously, this set is Borel (as the graph of a Borel function). According to the classical uniformization theorem of Luzin, Jankov and von Neumann (which is an easy consequence of the theorem of Kuratowski and RyllNardzewski), there exists a function
such that: 1) h is measurable with respect to the a-algebra generated by all analytic subsets of dom(h); 2) the graph of h is contained in 3. Further, our assumption on f and the definition of f * imply the relations: a) prx(dom(h)) = X ; b) for any x E X , we have (x, f * (3)) E dom(h). Consequently, we may write
Now, define a function g : X-+R by the formula g(x) = h(x, f "(t)
(3 E X I .
Then g is C-measurable (as a composition of two C-measurable functions) and we get f(.,g(x)) = f*(.) = inft,yf(x,t) for all points x E X .
Exercise 17. Let us consider the diagonal
234
C H A PTER
11
of the Euclidean plane R~ and let Z be a subset of this diagonal, nonme* surable with respect to the standard one-dimensional Lebesgue measure on it. Define a function f : R~ --+ R , putting f ( x , y) = 0 if (3, y) !$ Z , and f (x, y) = -1 if (x, y) E Z . Show that: a) for each point x R , the partial function f (a, is lower semicontinuous; b) for each point y E R , the partial function f(., y) is lower semicontinuous; c) the function 3 -, f ( x , 3) (x E R ) a)
is not Lebesgue measurable (consequently, f is not sup-measurable); d) f is measurable with respect to the usual two-dimensional Lebesgue measure on R 2 . Exercise 18. Suppose that there exists a Luzin subset of the Euclidean plane R ~ Demonstrate . that, in this case, there exists a mapping
satisfying the following relations: a) @ does not have the Baire property; b) for each function q5 : R -, R whose graph is a first category subset of the plane (in particular, for each 4 possessing the Baire property), the superposition 8 @(x,4 ( 4 ) (x E R ) possesses the Baire property. Deduce an analogous result (in terms of the Lebesgue measurability) from the existence of a Sierpiriski subset of the plane. +
Exercise 19. Let us denote by the symbol Mo = Mo[O,11 the family of all Lebesgue measurable functions acting from [O,l] into R . Obviously, we have a canonical equivalence relation E in Mo defined by the formula f
g
e f and g coincide almost everywhere on [O,l].
We denote by M the factor set with respect to this equivalence relation (i.e. M is the family of all equivalence classes with respect t o EE). If f is an arbitrary function from the original family Mo, then the symbol [f] will denote the class of all those functions which are equivalent to f .
SUP-MEASURABLE A N D W E A K L Y SUP-MEASURABLE FUNCTIONS
235
The natural algebraic operations in Mo are compatible with the relation s and, consequently, induce the corresponding algebraic operations in M . Therefore, M becomes a vector space over the field R. Further, for any two elements [f] E M and [g] E M , we put
Check that this definition is correct (i.e, it does not depend on the choice o f f and g) and that the function
obtained in this way turns out to be a metric on M . Show that: a) the pair (M, d) is a Polish topological vector space; b) there exists no nonzero linear continuous functional defined on the whole M (in other words, the conjugate space M * is trivial). Now, let us extend the equivalence relation introduced above onto the family of all Lebesgue measurable partial functions acting from [O, 11 into R. Namely, for any two such functions f and g, we put
f coincides with g almost everywhere on dom(f) n dom(g)). Further, we denote by the symbol M1 the family of all equivalence classes Evidently, we have the canonical embedding with respect to
=.
so we may identify M with the subset j ( M ) of MI. Suppose now that some mapping (operator)
is given such that: 1) for each element [f] E MI, we have the relation
where g is any Lebesgue measurable partial function for which [g] = H([f]); 2) if [f] E MI and f n (n < w) are the restrictions of f to any pairwise disjoint Lebesgue measurable subsets of R whose union coincides with dom(f 1, and H([fnI) = [gn] (n < w),
236
CHAPTER 11
then H([f]) = [g] where g is the common extension of all partial functions g, (n < w), for which dom(g)
= U{dom(g,)
: n
<w).
In this case, we say that H is an admissible operator acting from M' into itself. Under the assumption that the a-ideal I(X) of all Lebesgue measure zero subsets of R is c-additive (i.e. U{Xt : ( < P) E I ( A ) whenever card(P) < c and XC E I(X) for each 4 < p), demonstrate the existence of a mapping O : [0,1]x R - + R satisfying the following condition: for every Lebesgue measurable partial function f : [O, 11 R , +
the superposition
Of : [O, 11 -+ R is a Lebesgue measurable partial function, too, and the equality
holds true. In other words, any admissible operator H acting from M' into itself can be represented in the form of superposition operator (this result is essentially due to Krasnoselski and Pokrovski). Finally, formulate and prove an analogue of the preceding result for partial functions acting from [O,1] into R and having the Baire property (under the assumption that the a-ideal I i ( R ) of all first category subsets of R is c-additive).
12. Ordinary differential equations with bad
right-hand sides
In this chapter of the book we wish to consider some set-theoretical questions concerning the existence and uniqueness of solutions of ordinary differential equations. In particular, we deal here with those ordinary differential equations the right-hand sides of which are rather bad (even nonmeasurable in the Lebesgue sense), but for which we are able to establish the theorem on the existence and uniqueness of a solution. There are many simple examples of ordinary differential equations whose right-hand sides are not so good as their solutions. For instance, let us take the ordinary differential equation
Then it is easy to check that: 1) the right-hand side of this equation is continuous but not differentiable; 2) all solutions of this equation are analytic; 3) for any initial condition, there exists a unique solution of this equation, satisfying the condition. In our further considerations we shall show that some equations
are possible, for which f is nonmeasurable in the Lebesgue sense but relations 2) and 3) hold true. We begin with an old remarkable result of Orlicz (see [115]) stating that, for almost each (in the category sense) function O from the Banach space Ca(R x R ) consisting of all bounded continuous real-valued functions defined on R x R, the corresponding Cauchy problem
Y' = @(t, Y)
to) = Yo,
to
E R, yo E R )
238
CHAPTER
12
has a unique solution. In order to present this result we, first of all, want to recall the purely topological Kuratowski lemma on closed projections (see, e.g., Chapter 0 of the book). Namely, if X and Y are some topological spaces and, in addition, Y is quaaicompact, then the canonical projection
is a closed mapping, i.e, for each closed subset A of X x Y, the image prl(A) is closed in X . Several applications of the Kuratowski lemma were discussed in Chapter 0. Here we are going to present an application of this lemma to the theory of ordinary differential equations. Actually, we need here a slightly more general version of the lemma. Let us recall that a topological space E is a-quaaicompact if it can be represented in the form
where all sets E, (n < w) are quasicompact subspaces of E . Now, the following slight generalization of the Kuratowski lemma is true. Lemma 1. Let X be a topological space and let Y be a a-quasicompact space. Let, as above, prl denote the canonical projection from X x Y i n t o X . T h e n , f o r each F a -subset A of X x Y, the amage prl(A) i s a n Fa-subsei of
x.
Proof. In fact, the Kuratowski lemma easily implies this result. Indeed, since Y is a-quasicompact, we may write
where all Yn (n < w) are quasicompact subspaces of Y. Then, for any set A C X x Y, we have the equality p"i(A)
= U{prl(A n ( X
x Y,)) : n
< w),
Suppose now that A is an Fa-subset of X x Y. Then A can be represented in the form A = U{A, : m < w) where all sets A, (m
< w) are closed in X x Y. Therefore we obtain
prl(A) = U{prl(A,
n ( X x Y,))
: m
< w, n < w).
Now, every set A,n(XxYn)
(m<w, n<w)
ORDINARY DIFFERENTIAL EQUATIONS WITH B A D RIGHT-HAND SIDES
239
is closed in the product space X x Yn and every space Yn is quasicompact. Hence, by the Kuratowski lemma, the set
is closed in X . Consequently, prl (A) is an F,-subset of X . This completes the proof of Lemma 1. Now, let us return to the Banach space Cb(R x R ) . Then, for each function @ from this space, we can consider the ordinary differential equation
and, for any point ( t o , go) E R x R , we can investigate the corresponding Cauchy problem of finding a solution
of this equation, satisfying the initial condition ~ ( 8 0= ) Yo. It is well known (see, e.g., [119]) that such a solution does always exist and, since @ is bounded, the solution is global, i.e, it is defined on the whole real line R . On the other hand, we cannot assert, in general, the uniqueness of a solution. There are simple examples of continuous bounded real-valued functions @ on R x R for which the corresponding Cauchy problem admits at least two distinct solutions (in this connection, let us mention the famous work by Lavrentieff [95] where a much stronger result was obtained).
Exercise 1. Give an example of a function O belonging t o the space C b ( Rx R ) , for which there exists an initial condition (xo,go) E R x R such that the corresponding Cauchy problem
possesses at least two distinct solutions. Actually, we need some additional properties of the original function O in order to have the uniqueness of a solution of the differential equation
Y' = @(x,Y)
(y(xo) = go, co E R , go E R ) .
For instance, if @ satisfies the so-called local Lipschitz condition with respect to the second variable y, then we have a unique solution for each
240
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12
Cauchy problem corresponding to @. It is reasonable to recall here that satisfies the local Lipschitz condition with respect to y if, for any point ( t o ,yo) E R x R , there exist a neighbourhood V ( ( t o ,yo)) and a positive real number = M(@,(xo,Yo)), such that
<
I'(t, YI)- @(trY ~ ) J M lyl
- yzl
for all points ( t , yl) and (2, yz) belonging to V((x0, yo)). Let us denote by Lipl(R x R ) the family of all those functions from Cb(R x R ) which satisfy the local Lipschitz condition with respect to y. Then, obviously, Lipl(R x R ) is a vector subspace of C b ( R x R ) . Notice also that Lipl(R x R ) is a dense subset of C b ( R x R). Indeed, this fact is almost trivial from the geometrical point of view. Thus we can conclude that, for all functions belonging to some dense subset of C b ( R x R), the Cauchy problem Y'
= @(x,Y)
( ~ ( z o= ) yo, t o E R , yo E R )
has a unique solution. Orlicz essentially improved this result and showed that it holds true for almost all (in the category sense) functions from the Banach space Cb(R x R ) . More precisely, one can formulate the following statement.
Theorem 1. The set of all those functaons from C b ( R x R ) for which the corresponding Cauchy problem has a unique solutaon (for any point (20, pD) E R x R ) is a dense Ga-subset of C b ( R x R ) . Proof. Let us denote by the symbol U the family of all those functions from C b ( R x R ) for which the corresponding Cauchy problem has a unique solution (for each point ( t o, yo) belonging to R x R ) . As mentioned above, the set U is dense everywhere in C b ( R x R ) . Therefore it remains to prove that U is a Ga-subset of C b ( R x R ) . In order t o show this, let us first rewrite the Cauchy problem in the equivalent integral form: Y(X)=
1:
@(t,y(t))dt
+
Yo.
Further, for any two rational numbers E > 0 and q, let us denote by P ( E ,q) the set of all those elements ( a , t o , YO)E Ca(R x R ) x R x R
ORDINARY DIFFERENTIAL EQUATIONS WITH B A D RIGHT-HAND SIDES
241
for which there exist at least two real-valued continuous functions qil and 4z such that:
dom(41)= dom(qi2)= R,
rni(x) = 42(x) =
1: 1:
+
ro
(X
+
yo
(3 E R),
ect,m1ct))dt O(t,42(t))dt
E R),
It is not difficult to establish that P(E,q) is a closed subset of the product space Ca(R x R) x R x R. Indeed, suppose that a sequence
of elements of P(E,q) converges to some element
(@,xo, yo) E Ca(R x R) x R x R. Then we obviously have
and the sequence of functions
converges uniformly to the function O. We may assume without loss of generality that
(Vn E ~ ) ( l l @ ( ~ ) l l 5
11@11+
1).
and denote two real-valued conFor every natural number n, let tinuous functions satisfying the following relations:
&)(x) =
1'
xf'
d n ) ( t &)(t))dt ,
+
yp)
(x E R),
242
CHAPTER
l$ln)(q)
12
- 4P)(q)l L & -
Then it is not hard to verify that all functions from the family
are equicontinuous. More precisely, for each function $ from this family and for any two points x' E R and x" E R , we have the inequality
So, applying the classical Ascoli-Arzeld theorem (see, e.g., [119]), we can easily derive that there exists an infinite subset Ii' of N for which the partial sequences of functions
converge uniformly (on each bounded subinterval of R ) to some functions $1 and $2, respectively. Also, it can easily be checked that, for $1 and $ z , we have the analogous relations
Thus we see that ( @ I 80,YO )
E P ( & ,Q),
and hence P ( s ,q) is closed in the product space C b ( R x R ) x R x R . Now, let us put
P = u { P ( E , ~ ):
E
> 0,
C E Q , q E Q).
Then it is clear that a function E Cb(R x R ) does not belong to the set U if and only if there exist a rational number E > 0, a rational number q and some points xo E R and yo E R , such that (XQ, t o , yo) belongs to the set P ( E ,q). In other words, we may write
ORDINARY DIFFERENTIAL EQUATIONS WITH BAD RIGHT-HAND SIDES
243
where prl
:
Cb(R x R) x R x R + Cb(R x R)
denotes the canonical projection. It immediately follows from the definition of the set P that P is an Fa-subset of the product space Cb(Rx R) x R x R. In addition, the plane R x R is a a-compact space. So, applying Lemma 1, we conclude that prl(P) is an Fa-subset of Cb(R x R) and, consequently, U is a Ga-subset of Cb(R x R). This finishes the proof of Theorem 1.
Remark 1. Evidently, the Banach space Cb(Rx R) is not separable. Let E denote the subset of this space, consisting of all those functions which are constant at infinity. In other words, @ E E if and only if there exists a constant M = M ( @ )E R such that, for any for which we have
> 0, a
positive real number a = a(@,&)can be found
Notice that E is a closed vector subspace of Cb(R x R) and hence .E is a Banach space, as well. Moreover, one can easily verify that E is separable. Clearly, a direct analogue of Theorem 1 holds true for E. Actually, in [115] Orlicz deals with the space E. A number of analogues of Theorem 1, for other spaces similar to Ca(Rx R) or E, are discussed in [I].
Remark 2. Unfortunately, the set U considered above has a bad algebraic structure. In particular, U is not a subgroup of the additive group of Cb(Rx R) and, consequently, U is not a vector subspace of Cb(Rx R). Indeed, suppose for a while that U is a subgroup of Cb(Rx R), Then U must be a proper subgroup of Cb(Rx R). Let us take a function
Obviously,
U n ({s)+ U ) = 0.
+
But each of the sets U and {Q) U is the complement of a first category subset of Cb(Rx R). Therefore their intersection U n ({Q}+ U) must be the complement of a first category subset of Cb(R x R), too, and hence
We have thus obtained a contradiction which yields that U cannot be a subgroup of Cb(R x R).
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CHAPTER 12
For some other properties of U interesting from the set-theoretical and algebraic points of view, see e.g. [I]. Theorem 1 proved above shows us that, for many functions from the space Ca(Rx R ) , we have the existence and uniqueness of a solution of the Cauchy problem. In fact, this is one of the most important results in the theory of ordinary differential equations. Naturally, we may consider a more general class of functions
not necessarily continuous or Lebesgue measurable and investigate for such functions the corresponding Cauchy problem from the point of view of the existence and uniqueness of a solution. For this purpose, let us recall that, as shown in the previous chapter of our book, there exists a subset Z of the plane R x R , satisfying the following relations: 1) no three distinct points of Z belong to a straight line; 2) Z is the graph of some partial function acting from R into R; 3) Z is a Xz-thick subset of the plane R x R, where X z denotes the standard two-dimensional Lebesgue measure on R x R; 4) for any Bore1 mapping
the intersection of Z with the graph of d, has cardinality strictly less than the cardinality of the continuum. We denote by @ the characteristic function of the above-mentioned set 2 . Then, obviously, O is a Lebesgue nonmeasurable function and, furthermore, if [R]" C dom(X), then O is sup-measurable as well. Now, starting with the function O described above, we wish to consider an ordinary differential equation
with the Lebesgue nonmeasurable right-hand side 9, and we are going to show that, in some situations, it is possible to obtain the existence and uniqueness of a solution of this equation (for any initial condition). First of all, we need to determine the class of functions to which a solution must belong. It is natural to take the class ACl(R) consisting of all
ORDINARY DIFFERENTIAL EQUATIONS WITH B A D RIGHT- HAND SIDES
245
locally absolutely continuous real-valued functions on R. In other words, II, E ACr(R) if and only if, for each point x E R, there exists a neighbourhood V(x) such that the restriction $lV(x) is absolutely continuous. Another characterization of locally absolutely continuous functions on R is the following one: a function $ belongs to ACl(R) if and only if there exists a Lebesgue measurable function
such that f is locally integrable and
for any point x E R. Let ly be a mapping from R x R into R and let (30,yo) E R x R. We say that the corresponding Cauchy problem
has a unique solution (in the class ACl(R)) if there exists a unique function $ E AC,(R) satisfying the relations: a) $'(x) = ly(x, $(x)) for almost all (with respect to the Lebesgue measure A) points x E R; b) $(go) = Yo. For example, if our mapping ly is bounded, Lebesgue measurable with respect to x and satisfies locally the Lipschitz condition with respect to y, then, for each (xo,YO)E R x R, the corresponding Cauchy problem has a unique solution. The reader can easily verify this fact by using the standard argument. Notice that, in this example, ly is necessarily Lebesgue measurable and sup-measurable (cf. Exercise 2 from Chapter 11). Notice also that an analogue of Theorem 1 holds true for a certain class of Banach spaces consisting of mappings (acting from R x R into R ) which are Lebesgue measurable with respect to x and continuous with respect to y.
Exercise 2. Prove that an analogue of Theorem 1 remains true for any Banach space E of bounded mappings acting from R x R into R, for which there exists an everywhere dense set D 2 E such that each function from D is Lebesgue measurable with respect to x and satisfies locally the Lipschitz condition with respect to y. The next statement shows that the existence and uniqueness of a solution can be fulfilled even for some ordinary differential equations whose
246
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12
right-hand sides are extremely bad, e.g. nonmeasurable in the Lebesgue sense.
Theorem 2. There is a Lebesgue nonmeasurable mapping
such that the Cauchy problem
has a unique solution for any point (xD,yo) E R x R. Proof. Let Z be a subset of the plane, constructed in the previous chapter (see Theorem 2 therein). Denote again by the characteristic function of Z and fix a real number t . Further, put
We assert that ly is the required mapping. Indeed, is Lebesgue nonmeasurable because @ is Lebesgue nonmeasurable. Let now (30,yo) be an arbitrary point of the plane R x R. Consider a function
defined by the formula
The graph of this function is a straight line, so it has at most two common points with the set 2. Consequently, the function
is equal to t for almost all (with respect to the Lebesgue measure A) points from R. We also have $'(x) = t for all x E R. In other words, $ is a solution of the Cauchy problem
It remains to show that $ is a unique solution from the class ACl(R). For this purpose, let us take an arbitrary solution 4 of the same Cauchy problem, belonging to ACl(R). Then, for almost all points x E R, we have the equality 4'(.) = @ ( X I 4(.>) t.
+
ORDINARY DIFFERENTIAL EQUATIONS WITH BAD RIGHT-HAND SIDES
247
It immediately follows from this equality that the function @+ is measurable in the Lebesgue sense. But, as we know,
So we obtain that @,p is equivalent to zero and hence
for almost all x E R. Therefore we can conclude that
This completes the proof of Theorem 2.
Remark 3. The preceding theorem was proved in the theory ZFC.In this connection, let us stress once more that the function ly of Theorem 2 is Lebesgue nonmeasurable and, under a certain set-theoretical hypothesis, is also sup-measurable (hence weakly sup-measurable). At the same time, we do not know whether it is possible to establish within the theory ZFC the existence of a sup-measurable mapping which is not measurable in the Lebesgue sense. Exercise 3. Let n be a natural number and let
be a polynomial of degree n. Show that there exists a mapping
satisfying the following relations: a) ly is nonmeasurable in the Lebesgue sense; b) for any initial condition (xD,yo) E R x R, the differential equation y' = ly(x, y) has a unique solution $J with $J(xo)= yo; c) all solutions $J of the above-mentioned differential equation are of the form $(x) = aoxn alxn-I ... an,lx a (x E R ) ,
+
+ +
+
where a E R. (constructed in the previous Now, starting with the same function chapter of the book), we shall show that, under the set-theoretical assumpt ion [R]" c cdom(X),
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CHAPTER
12
Theorem 1 of Orlicz can be generalized to Banach spaces of mappings acting from R XR into R, essentially larger than the classical Banach space C ~ ( R X R) (notice that all spaces of real-valued bounded mappings, considered in this chapter, are assumed to be equipped with the norm of uniform convergence). More precisely, we can formulate and prove the next result.
Theorem 3. Suppose that [R]
has a unique solution for every anWial condition y(xo) = yo, is a dense Ga-subset of Bo. be again a mapping constructed in the previous Proof. Let @ = chapter of the book. Obviously, this mapping does not belong to the vector space Ca(Rx R). Denote by Bo the vector space of functions, generated by {@o)U Cb(R x R). Clearly, each function q belonging to Bo can be represented in the form
E Cb(R x R) and t l E R. Moreover, since Bo is the direct sum where of the vector spaces Cb(R x R) and {tGo : t E R), such a representation is unique. We equip Bo with the norm of uniform convergence. Taking account of the fact that @o is Lebesgue nonmeasurable, we may write
dist(@o,Cb(R x R))
> 0.
In other words, Bo can be regarded as a topological direct sum of the two Banach spaces Cb(R x R) and {tao : t E R). Consequently, we may identify Bo with the product space Cb(R x R) x R. Let now 91 be an arbitrary function from Cb(Rx R) such that the corresponding ordinary differential equation
ORDINARY DIFFERENTIAL EQUATIONS WITH BAD RIGHT-HAND SIDES
249
has a unique solution for any initial condition y(xo) = yo. Then it is not difficult t o check (by using the properties of our function Qo) that, for each real number t l , the ordinary differential equation
has also a unique solution for any initial condition y(xo) = yo. Conversely, if a function ly = ly, +tl@,, from the space Bo (where q1 E C b ( R x R ) ) is such that the ordinary differential equation Y' = Q(x1 possesses a unique solution for every initial condition, then the ordinary differential equation Y' = *1(x,y) possesses a unique solution for every initial condition, too. Let us recall that the symbol U denotes (in this chapter) the family of all functions l y l from Cb(R x R ) such that the differential equation y' = g l ( x , y) has a unique solution for any initial condition. Denote now by V an analogous family for the space Bo, i.e. let V be the family of all functions ly from Bo such that the differential equation y' = ly(x, y) has a unique solution for any initial condition. Then, taking account of the preceding argument, we can assert that
V = U + {tQ0 : t E R ) . Since, according t o Theorem 1, U is a dense Gs-subset of the Banach space C b ( R x R ) , we easily conclude that V is a dense Gs-subset of the Banach space Bo . Theorem 3 has thus been proved. Exercise 4. By assuming the same hypothesis
give an example of a Banach space B1 of functions acting from R x R into R, satisfying the following relations: 1) Cb(R x R ) C B1; 2) there are discontinuous Lebesgue measurable sup-measurable functions belonging to B1; 3) there are Lebesgue nonmeasurable sup-measurable functions belonging to B1;
250 4) an analogue of Theorem 1 holds true for
C HA PTER
12
B1.
Remark 4. Let B be a Banach space of bounded sup-measurable mappings, for which an analogue of Theorem 1 is valid, i.e, the family of all Q E B such that the differential equation
has a unique solution for every initial condition y(xo) = yo, is a dense Gasubset of B. It is not difficult to see that the class of all Banach spaces B is rather wide. In particular, it follows from Theorem 3 that the situation is possible where a space of this class contains a Lebesgue nonmeasurable mapping. In this connection, it would be interesting to obtain a characterization (description) of the above-mentioned class of Banach spaces. Finally, let us point out that some logical and set-theoretical aspects of the classical Cauchy-Peano theorem on the existence of solutions of ordinary differential equations are discussed in the paper by Simpson [145].
13. Nondifferentiable functions from the point of view of category and measure
Earlier we were concerned with various nondifferentiable functions acting from R into R. In this chapter, we wish to discuss one general approach to such functions from the viewpoint of category and measure. Roughly speaking, our goal is to demonstrate that, for a given generalized notion of derivative (introduced within the theory ZF & DC), the set of nondifferentiable functions (with respect to this notion) turns out t o be sufficiently large. We begin with an approach based on the concept of Baire category. More precisely, it is based on the important theorem of Kuratowski and Ulam from general topology (for the formulation and proof of this theorem see, e.g., [89] or [117]). Note that the Kuratowski-Ulam theorem can be interpreted as a purely topological analogue of the classical Fubini theorem from measure theory. It is widely known that the Fubini theorem is fundamental for all of measure theory. Moreover, this theorem has many applications in analysis, probability theory and other domains of mathematics. Also, it is well known that the Kuratowski-Ulam theorem possesses a number of nontrivial applications in general topology and in modern mathematical analysis (some of them are presented in the books 1891 and [117]). In our further considerations, the main role is played by the following statement.
Theorem 1. Let El and Ez be a n y t w o topological spaces with countable bases ( or, more g e n e ~ a l l y ,with countable P - bases ) and let E3 be a topological space. Let Z be a subset of the product space El x E z . Suppose that a certain mapping @:Z-+E3 i s given, and that this mapping satisfies the conditions: 1) the partial function @ acting f r o m the topological space El x E2 i n t o the topological space E3 has the Baire property, i.e., f o r a n y open set V
from E3, the preimage W1(V) has the Baire property an El x Ez; 2) for almost all (in the sense of category) points x E El, the domaan of the ~artialmapping @(x,.) given b y
is a first category set in the space Ep. Then the following two relations hold: (a) Z is a first category subset of the product space El x E2; (b) for almost all (in the sense of category) points y E E2, the set
as of first category in the space El; roughly speaking, almost each point y E El is almost singular with respect to the partial mapping @(.,y). The proof of this general statement is very simple. Indeed, according to the Kuratowski-Ulam theorem, relation (a) implies relation (b). Therefore it is sufficient to establish relation (a) only. Since, in virtue of condition I), the partial function @ has the Baire property, the set
has the Baire property in the product space El x Ep. Using condition 2) and the Kuratowski-Ulam theorem once more, we get the required result. In connection with Theorem 1, a natural question arises: how can condition 2) be checked for the given partial mapping @? The following situation can be frequently met in analysis and it will be the most interesting for us in the sequel. Suppose that Ez is a Polish topological vector space, E3 is a topological vector space with a countable base and our partial mapping @ satisfies condition 1) and the next condition: 2') for almost each (in the sense of category) point x E E l , the partial mapping @(x,.) is linear and discontinuous on its domain. Then it can be shown that @ satisfies condition 2), as well. Indeed, for almost all points x E El, the function @(x,.) has the Baire property and is linear and discontinuous on the vector space
Let us prove that, for the points x mentioned above, the set Z(x) is of first category in the space Ep. Suppose otherwise, i.e, suppose that Z(x) is a second category set with the Baire property. Then we may apply to Z(x) the well-known Banach-Kuratowski-Pettis theorem from the theory of
253
N O N DIFFE R E N T I A BLE F U N C TI O N S
topological groups (see, for example, [66] or [89]). This theorem is a topological analogue of the classical Steinhaus property of Lebesgue measurable sets with a strictly positive measure. Namely, according to this theorem, the set Z(x) Z(x) = { y - 2 : y E Z(x), 2 E Z(x))
-
contains a nonempty open subset of the topological vector space Ez (more precisely, the set Z(x) - Z(x) is a neighbourhood of zero of Ez). But since the set Z(x) is a vector space, too, we come to the equality
and, finally, we obtain Z(X) = Ez. Hence the function @(x,.) is defined on the whole Polish topological vector space Ez and is linear on this space. Now, by taking account of the fact that the function @(x,.) has the Baire property, it is not difficult to prove (by using the same Banach-Kuratowski-Pettis theorem) that @(x,.) is a continuous mapping. But this contradicts the choice of the point x. The contradiction obtained shows us that the set Z(x) must be of first category in the space E2,Therefore condition 2) is satisfied for our partial mapping
a. Remark 1. Theorem 1 may be considered as one of possible formalizations of a well-known principle of mathematical analysis which is frequently called "the principle of condensation of singularities". Among various works devoted to this principle, the most famous is the classical paper of Banach and Steinhaus [8]. It is easy to see that the Banach-Steinhaus principle of condensation of singularities is closely connected with Theorem 1 and can also be obtained as a consequence of the Kuratowski-Ulam theorem. Indeed, let us take El = N where the set N of all natural numbers is equipped with the discrete topology, and let E2 be an arbitrary Banach space. Suppose that E g is another Banach space and a double sequence of continuous linear operators
Lm,, : E2 -+ E3
( m ,n E N)
is given, such that, for any m E N , we have
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13
Let us define a partial mapping from the product space El x Ez into the space E3 by the following formula:
is clear that this partial mapping has the Baire property and, for each E N , the partial mapping @(m,.) is defined on a first category subset the space E z . Hence the domain of the partial mapping @ is also a first category set in the product space El x E z . Now, we may apply the Kuratowski-Ulam theorem and, evidently, we obtain that, for almost all elements x E El, the set { m : (m, x ) E dam(@))
is empty. But, actually, this is the Banach-Steinhaus principle of condensation of singularities.
Remark 2. The general scheme of applications of Theorem 1 is as follows. First of all, we must check that a given partial mapping @ has the Baire property. Obviously, @ has this property if it is a Bore1 mapping or, more generally, if it is a measurable mapping with respect t o the aalgebra generated by a family of analytic sets (such situations are typical in modern analysis). Now, suppose that our partial mapping @ of two variables has the Baire property. Then the second step is to check that the corresponding partial mappings of one variable are defined on the first category sets. This will be valid if Ez and E3 are Polish topological vector spaces and if, for almost all elements x E E l , the corresponding mappings @(x,.) are linear and discontinuous on their domains (notice that if the given space Eg is a normed vector space, then we need to check the linearity and the unboundedness of the corresponding partial mappings). Finally, we can apply Theorem 1. Now, we wish to present an application of Theorem 1 in a concrete situation. Namely, we will be interested in a certain type of generalized derivative. Let co denote the separable Banach space consisting of all real-valued sequences converging to zero. Let R denote the real line and let [O,l] be the closed unit interval in R. Suppose that a mapping
is given. Evidently, we may write
NONDIFFERENTIABLE FUNCTIONS
where 4n:[0,1]-,R
EN).
Let us assume that the mapping 4 satisfies the following condition: for each point x E [O,11 and for each index n E N, the value &(x) is not equal to zero. Moreover, let us assume (without loss of generality) that
for all natural numbers n. If f is a real-valued function defined on the segment [0, 11 and a point x belongs to this segment, then the real number
is called the +derivative of f at x (if this limit exists, of course). In our further considerations, we denote the limit mentioned above by the symbol f$(x>. Let us put El = [O, 11, Ez = C[O, 11, E3 = R and consider a partial mapping acting from the product space El x E2 into the space E3 and defined by the formula
Suppose that the original function 4 has the Baire property. We assert that, in such a case, the partial mapping has the Baire property, too. Indeed, it suffices to observe that, for every natural number n, the mappings
have the Baire property. For the second mapping, this is obvious since the function 4% has the Baire property. Further, the mapping
is continuous and the mapping
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C H A PTER
13
can be represented as the following superposition:
In this superposition the first mapping has the Baire property and the two other mappings are continuous. Therefore we conclude that the superposition also has the Baire property. Let us notice, by the way, that the same result can be established in a different manner. Namely, if in the function of two variables (31f ) f(x dln(x)) f (XI
+
+
-
we fix a point x , then we obtain a continuous function of one variable, and if in the same function of two variables we fix a second variable f , then we obtain a function of one variable having the Baire property. So, we see that, for our function of two variables, the conditions similar to the classical Caratheodory conditions (i.e, the measurability with respect to one of the variables and the continuity with respect t o another one) are fulfilled. From this fact it immediately follows that our function of two variables has the Baire property (in this connection, see also [lo21 where a general problem concerning the measurability of functions of two or more variables is investigated in detail). Taking the above remarks into account, we conclude that the partial mapping (2, f fk<x> has the Baire property. Moreover, it is easy to see that if a point x is fixed, then this partial mapping yields a linear discontinuous function of one variable f. Consequently, we can apply Theorem 1 and formulate the following statement.
>
-
Theorem 2. If a mapping
has the Baire property, then almost each function from the Banach space C[O, 11 does not possess a 4-derivative almost everywhere on the segment
LO, 11. We want to point out that the basic operations used in classical mathematical analysis are, as a rule, of the projective type, i.e, these operations are described completely by some projective sets lying in certain Polish topological spaces. In many natural situations, it can happen that the graph of our partial mapping from Theorem 1 is a projective subset of the corresponding Polish product space. Then, according to the important
N O N D I FFE R E N T I A B LE F U N C TI O N S
257
results of Solovay, Martin and others, we must apply some additional settheoretical axioms for the validity of the corresponding version of Theorem 1. For example, suppose that satisfies only condition 2) of Theorem 1, the graph of lies in a Polish product space El x E2 x E3 and this graph is a continuous image of the complement of an analytic subset of a Polish topological space. Then if we wish to preserve the assertion of Theorem 1 for a, we need the existence of a two-valued measurable cardinal or Martin's Axiom with the negation of the Continuum Hypothesis. Analogously, if the graph of our partial mapping is a projective subset of a Polish product space, belonging to a higher projective class, then we need the Axiom of Projective Determinacy or a similar set-theoretical axiom (for more details, see [55] and [56]). Actually, suppose that we work in the following theory: ZF & D C & (each subset of R has the Baire property). Then the assertion of Theorem 1 will be true for all Polish topological spaces El, Ez, Es and for all partial mappings acting from El x E2 into E3 and satisfying condition 2) of this theorem. See, e.g., [67] where the theory mentioned above is applied to some questions connected with the existence of generalized derivatives of various types. In particular, it is established in 1671 that if we work in the above-mentioned theory, then almost each function from the space C[O, 11 does not possess a generalized derivative almost everywhere on the segment [O, 11. Obviously, such an approach can also be applied to special types of generalized derivatives, for instance, to the so-called path derivatives (for the definition and basic properties of path derivatives, see, e.g., [21]). In addition, let us stress that the direct analogue of the classical BanachMazurkiewicz theorem (which was considered in Chapter 0) cannot be established for all generalized derivatives, since there is (in the theory ZF & D C ) a certain notion of a generalized derivative having the property that, for any continuous function there exists at least one point x from the segment [O, 11, such that f is differentiable at x in the sense of this generalized derivative (cf. [67]). Further, the following natural question arises: does there exist an an* logue of the above-mentioned result in terms of measure theory'? In other words, does there exist a Bore1 diffused probability measure p on the space C[O, 11 such that, for any generalized derivative introduced in the theory ZF & D C , almost all (with respect to p ) functions from C[O, 11 are not differentiable, in the sense of this derivative, at almost all (with respect to A) points of [0, l]?
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13
At the present time, this question remains open. Here we give a construction of the classical Wiener measure p, on C[O, 11 and demonstrate that, for the derivative in the usual sense, p , yields a positive answer to this question. We recall that historically the Wiener measure appeared as a certain interpretation (mathematical model) of the Brownian motion (for an interesting survey of this phenomenon, see e.g. [14] and, especially, [96]). Note that the construction of the Wiener measure is not easy and needs a number of auxiliary facts and statements. To begin, we first of all wish to recall some simple notions from probability theory and the famous Kolmogorov theorem on mutually consistent finite-dimensional probability distributions. Let E be a set, let S be a a-algebra of subsets of E and let p be a probability measure on S . So we are dealing with the basic probability space (ElS , ~ 1 ) In our further considerations we assume, as a rule, that p is a complete measure. Let f be a partial function acting from E into R. We say that f is a random variable i f f is measurable with respect to the a-algebra S (i.e, for any open set U C_ R, the preimage f"(U) belongs to S ) and
-
For any random variable f , we may define the Bore1 probability measure p j on R, putting
The measure p j is usually called the distribution of a random variable f . Actually, the measure p j is defined in such a way that it becomes the homomorphic image of the measure p under the homomorphism f , so we may write p j = p o f - l . Obviously, p~ is uniquely determined by the function Fj : R [O,11 such that
F j ( x ) = p ( { e ~ E: f ( e ) < x ) ) (xER). This function is also called the distribution o f f . It is increasing and satisfies the relations: (a) limt,-o,Fj(t) = 0,
NONDIFFERENTIABLE FUNCTIONS
(b) limt,+, Ff (t) = 1, (c) (Vx E R)(limt,$- Ff (t) = Ff (x)).
Exercise 1. Let F be an increasing function acting from R into R and satisfying the relations analogous to (a), (b) and (c). Show that there exist a probability space (E,S, p) and a random variable
such that F = Fj.
Exercise 2. We recall that a probability measure p is separable if the topological weight of the metric space canonically associated with p is less than or equal to w (in other words, the above-mentioned metric space is separable). For instance, the classical Lebesgue measure on the unit segment [0, 11 is separable. Check that this fact is a trivial consequence of the following statement: the completion of any probability measure given on a countably generated a-algebra of sets is separable. Show the validity of this statement. Check that any homomorphic image of a separable measure is separable, too. Give an example of a topological space T and of a Borel probability measure on T which is not separable. Remark 3. In connection with the result of Exercise 2, let us note that there exist nonseparable measures on the segment [0, 11 extending the classical Lebesgue measure on [O,l]. Moreover, there are nonseparable extensions of the standard Lebesgue measure on the unit circle, which are invariant under the group of all rotations of this circle around its centre (for more information, see e.g. [57], [69] and references therein). Exercise 3. Let (E,S,,u) be a basic probability space and let T be a topological space (equipped with its Borel a-algebra B(T)).Any pmeasurable partial mapping
satisfying the condition
is usually called a T-valued random variable on E. The Borel probability measure p j on T defined by the formula
is called the distribution of f in T, and we write p j = ,u o f - l . Show that there exist a probability space ( E , S , p ) , a topological space T and a Borel probability measure v on T, such that there is no T-valued random variable f on E for which pf = v . Let ( E l S , p ) be again a basic probability space and let
be a random variable. We recall that JE f(e)dp(e) is the mathematical expectation of f (of course, under the assumption that this integral exists). We also recall the simple formula
More generally, for any Borel function
we have
under the assumption that the corresponding integrals exist. Exercise 4. Prove the formula presented above. Deduce, in particular, that, for each natural number n, the equality
holds true (if these integrals exist). In many cases, it may happen that the distribution p j of a random variable f can be defined with the aid of its density. We recall that a Lebesgue measurable function
is a density of p j (of F j ) if, for each Borel set X C_ R, we have
NONDIFFERENTIABLE FUNCTIONS
26 1
This means that the measure p j is absolutely continuous with respect to the Lebesgue measure X on R. Evidently, any two densities of p j are equivalent with respect to A. In addition, if p j exists, then we can write
and, more generally,
for every Bore1 function q5 :
R-+R
such that the corresponding integrals exist. The classical example of a probability distribution is the normal (or Gaussian) distribution. For the real line, the density of the so-called centered normal distribution is given by the formula
where a
> 0 is a fixed constant.
It can easily be checked in this case that
Taking a derivative (with respect to a ) in the last equality, we obtain
where c is some strictly positive constant whose precise value is not interesting for us. We now wish to recall the Kolmogorov theorem on the existence of a probability measure with given finite-dimensional distributions (see, e.g., [14], [36], [113], [121]). This theorem plays the fundamental role in the contemporary theory of stochastic processes. Let T be an arbitrary set of indices. Consider a family {& : t E T ) where, for each index t E T, the set & coincides with R. Suppose that, for any finite set T = { t l , ..., tn) T,
a Borel probability measure p, on the space
is given in such a way that the whole family
{pr :
7
E
[WW)
of probability measures is consistent, i.e, for any two finite subsets T' of T such that T E r', we have
T
and
where
p r T ~ , :, RTj + RT denotes the canonical projection from R,, onto R,. Further, consider the product space
with the a-algebra S generated by the family of mappings
where, for each index t E T , the mapping
coincides with the canonical projection from RT onto Rt. In other words, we may say that S is the smallest a-algebra of subsets of RT,such that all mappings prt (t E T) are measurable with respect to S ( S is also frequently called the cylindrical a-algebra of the space RT). Exercise 5. Show that the cylindrical a-algebra of the topological R coincides with its Borel a-algebra if and only if product space '
Exercise 6. Let X be a set and let { f i : i E I) be a family of realvalued functions defined on X. We say that this family separates the points of X if, for any two distinct points x and y from X , there exists an index i E I such that
fa(x> 2 fa(!/).
N O N DIFFERE N TI A BLE F U N C TI O N S
263
Let now X be a Polish topological space and let {fd : i E I ) be a countable family of Borel real-valued functions on X , separating the points of X . Denote by S({fi : i E I ) ) the smallest a-algebra of subsets of X , for which all functions fr (i E I ) become measurable. Consider a mapping
defined by the formula
Note that, since carcd(I) 5 w , the space RI is isomorphic to one of the spaces R W , R n (n E N). Check that: a) f is injective and Borel; b) S({fi : i E I)) = {f-l(Z) : Z E B(R')). By using the classical theorem from descriptive set theory, stating that the image of a Borel subset of a Polish space under an injective Borel mapping into a Polish space is also Borel, infer from a) and b) the equality S({fi : i E I ) ) = B(X). In particular, consider the separable Banach space C[O, 11 of all continuous real-valued functions on the segment [0, 11 and take as I a countable subset of [0, 11 everywhere dense in [0, I]. For each i E I, let
be the mapping defined by
Conclude from the result presented above that S({fi : i E I ) ) = B(C[O, 11). Give also a direct proof of this equality, without the aid of the mentioned result. The Kolmogorov theorem states that there exists a unique probability measure p~ defined on the cylindrical a-algebra of RT and satisfying the relations (r E Clr = PT 0 PTT,;
CHAPTER 1 3
where, for each finite set r 5 T, the mapping
is the canonical projection from RT onto R,. The original measures p, are usually called the finite-dimensional distributions of p ~ . The proof of the Kolmogorov theorem is not difficult. Indeed, using the consistency conditions, we first define the functional p~ on the cylindrical algebra (consisting of all finite unions of elementary subsets of R ~ in) such a way that the equalities Pr = pT P~T,: will be fulfilled for all finite sets r C T . Then we have to show that this functional is countably additive on the above-mentioned algebra. This is not hard because all finite-dimensional spaces R, are Radon, i.e, for any Bore1 set X C RT and for each E > 0, there exists a compact set K 2 X such that p r ( X \ K) < E . Finally, utilizing the classical Carathkodory theorem, we can extend our functional onto the whole cylindrical a-algebra S (for details, see e.g. [14], [I131 or [121]). Exercise 7. With the previous notation, show that, in the formulation of the Kolmogorov theorem, it suffices to assume only the consistency conditions of the form -1 PT = PT' 0 Prrl,r, where r and r' are any finite subsets of T for which T
c r',
carcd(rl \ r ) = 1.
R e m a r k 4. There are various generalizations of the Kolmogorov theorem. For example, this theorem may be regarded as a particular case of the statement asserting the existence of a projective limit of a given projective system of Radon probability measures. Furthermore, there are some abstract versions of the Kolmogorov theorem in terms of the so-called compact classes of sets. For more details, see e.g. [14], [113] or [121]. For our further purposes, we need only that case of the Kolmogorov theorem when T = [O,l].
NONDIFFERENTIABLE FUNCTIONS
Let us fix a finite set
= { t l l *..,tn) c [011]\ ( 0 ) . Clearly, we may suppose that 0 < t l measure p, on R, by the formula
< ... < t n .
Define a Borel probability
where the density p, satisfies the relation
e~p((-1/2)(x:/tl for all points
+(
-
~ 2~
l ) ~ / (-t tal ) ( ~ 1 s1s
s , ~ n )
+ ... + ( x n - ~ n - ~ ) ~ /-( ttnn- I ) ) )
E
If T is a finite subset of [0,11 whose minimal element coincides with 0 , then we put P7 = Po x C17\{0) where po is the Borel probability measure on Ro concentrated a t the origin of Ro (i.e. the so-called Dirac measure). It is not difficult to check the consistency of the family of probability measures
{p, :
T
i s a f i n i t e subset of [ O l l ] ) .
Exercise 8. By starting with the equality
ItW ex p (- ax /2)dx = ( 2 r / a ) ' f 2 2
( a > 0),,
show that
= ( 2 n c d / ( c+ $))'la - exp((-1/2)((a - b)'/(c + d ) ) ) , where a , b, c, d are strictly positive real numbers. Exercise 9. By using the result of Exercise 8, demonstrate the consistency of the above-mentioned family of measures
{p, :
T
i s a finite subset o f [ O , l ] ) .
266
C H A P T ER 13
Applying the Kolmogorov theorem to this family of measures, we get the probability measure p, on the product space We shall demonstrate below that the latter measure canonically induces the required Wiener measure on the space C[O, l ] C (in this connection, note that the initial measure p, also is called the Wiener measure on the product space ~['l']). In order to obtain the main result of this section, we need some simple but important notions from the general theory of stochastic processes. Let (E,S, p ) be a space endowed with a probability measure and let T be a set of indices (parameters). We shall say that a partial function of two variables H : ExT-tR ~ [ ' l l ] .
~ [ ' l l ]
is a stochastic process if, for each t E T, the partial function is a random variable on the basic probability space ( E , S, p ) . In this case, for any fixed e E E, the partial function H(e,.) : T - R is called the trajectory of a given process H , corresponding to e. Suppose that T is equipped with a a-algebra S' of its subsets, i.e, the pair (T, S') turns out to be a measurable space. We say that a stochastic process H is measurable if it (regarded as a partial function on E x T) is measurable with respect to the product a-algebra of S and S'. Exercise 10. Let us put E = T = [O, 11 and equip [O,1] with the standard Lebesgue measure A . Give an example of a nonmeasurable stochastic process H such that dom(H) = E x T and all trajectories H(e, .) (e E E ) and all random variables H(., t) (t E T ) belong to the first Baire class. Suppose that some two stochastic processes H and G are given on E x T. We say that they are stochastically equivalent if, for each t E T, the random variables H(., t ) and G(., t) are equivalent (i.e, coincide almost everywhere with respect to p ) . Stochastically equivalent processes have very similar properties and, as a rule, are identified. However, in certain problems of probability theory (e.g. in those where special features of trajectories of a given process play an essential role) such an identification cannot be done. Assume now that a set T of parameters is a topological space. We say that a stochastic process H : ExT-tR
267
N O N D I F F ER E N T I A B L E F U N C T I O N S
is stochastically continuous at a point to E T if, for each
E
> 0, we have
Further, we say that a process H is stochastically continuous if H is stochastically continuous at all points t E T . Note that if H I and H z are any two stochastically equivalent processes, then H I is stochastically continuous if and only if H z is stochastically continuous. Exercise 11. Suppose that the unit segment [0, 11 is equipped with the Lebesgue measure A . Give an example of a measurable stochastic process H with d o m ( H ) = [O,11 x [0, I], which is stochastically continuous but almost all its trajectories are discontinuous.
Lemma 1. Let T = [O,l] wath the usual topology and let
be a stochastic process. Then the following two conditions are equivalent: I ) H is stochastacally continuous; 2) for any E > 0, we have
Proof. Suppose that condition 1) is fulfilled. Fix E > 0 and 6 > 0. For each t E T, there exists an open neighbourhood V ( t ) o f t such that
The family { V ( t ) : t E T ) forms an open covering of T = [0,1]. Since [O, 11 is compact, there exists a Lebesgue number d > 0 for this covering, i.e, d has the property that any subinterval of [O,I] with diameter 2d is contained in one of the sets of the covering. Consequently, if
then t 1 E It
- d , t + d[ and, for some r E T , we get
Thus, for almost all e E E, we may write
268
CHAPTER 13
{e : IH(e,tl)
- H ( e , r ) J> &/2) U {e
:
J H ( e , t )- H(e,y)I > e/2)
and, taking into account the definition of V ( r ) ,we obtain
This establishes implication 1) j 2). The converse implication 2) trivial, and the lemma has thus been proved.
1) is
Exercise 12. Show that Lemma 1 holds true in a more general situation when T is an arbitrary compact metric space. Exercise 13. Let T = [O, 11 and let
be a stochastic process. Suppose also that, for some real number cu there exists a function
4
>
0,
: [O, 11 -t [O, +oo[
satisfying the following two conditions: 1) limd,o+d(d) = 0; 2) for all t and t' from [O,l], we have
Show that the process H is stochastically continuous. The simple result presented in Exercise 13 can directly be applied to the Wiener measure pw introduced above. Indeed, we have the basic probability space (R[OJl, S,pw) and the stochastic process W :
~
[
~
l
'
x [0, 11 -t R ]
canonically associated with p,, which is defined by the formula
In particular, we see that dom(W) = R[OI'] t l and tz from [O,1] such that
x [O,l]. Choose any two points
269
N O N DIF F ERE N T I A BL E F U N C T I O N S
According to the definition of p,, the two-dimensional distribution of the random vector (W(.,tl), W ( ~ l t 2 ) ) is given by the corresponding density
where, for all (xl, x2) E R 2 , we have Pt1,ta( X I , X Z= ) (1/2n)(tl(t2
- t1))-'/~ezp((-1/2)(x~/tl + (22 - ~ 1 ) ~ / (-t 2tl))).
Consider the random variable W(., t l ) - W(*,t2)e It is easy to see that the density p : R-tR
.
of this variable is defined by the formula p(x) = (2n(t2 - tl))'1/2exp(-x2/2(t2
- tl))
(x E R ) .
Indeed, this immediately follows from the general fact stating that if (fl, f2) is a random vector whose density of distribution is q(f1 , f 2 ) : R 2
then the density of distribution of fi -
+
f2
R,
is
where qfi-,a(x) = JR P(fl,h)(X+ I.I)d8
(x E R ) .
Exercise 14. Prove the fact mentioned above. Now, if t and t' are any two points from [O,l], we may write p,({e
-
E R['I~]: (W(e,t) W(e,t1)l > E ) )
<
270
CHAPTER 1 3
This shows us that the process W is stochastically continuous. W is usually called the standard Wiener process. Let us remark that W may be regarded as a canonical example of a Gaussian process (for information about Gaussian processes, see e.g. [36], [I131 and [121]). Let us return to a general probability space ( E , S, p) and assume that T is a set of parameters equipped with some u-algebra S' of its subsets. Consider two stochastic processes
H : ExT-tR,
G : ExT-tR.
We shall say that G is a mea'surable modification of H if the following conditions are fulfilled: a) H and G are stochastically equivalent; b) G is a measurable process, i.e. G regarded as a partial function acting from E x T into R is measurable with respect to the product u-algebra of S and S'. In particular, if (T, S', v) is a probability space and G is a measurable modification of H, then we also say that G is a ( p x v)-measurable modification of H. But, sometimes, it is more convenient to define a (p x v)-measurable modification of H as a stochastically equivalent process measurable with respect to the completion of the product measure p x v. Suppose that our set T of parameters is a topological space. We shall say that a stochastic process
is separable if there are a p-measure zero set A C E and a countable set Q 2 T, such that, for any element e E E \ A and for any point t E dom(H(e, -)), there exists a sequence
{t,
: n E N}
C Q n dom(H(e,
)s
converging to t and having the property
From this definition follows at once that Q is dense in T , so T is separable (as a topological space). The above-mentioned set Q is usually called a set of separability of H.
27 1
NONDIFFERENTIABLE FUNCTIONS
Lemma 2. Let T coincide with the unit segment [O,1] equipped with the standard Lebesgue measure A, and let
be an arbatrary stochastically continuous process. Then there exists a process
satisfying the relations: 1) H and G are stochastically equivalent; 2) G is measurable; 3) G is separable and one of its sets of separability coincides with
4) there exists a p-measurable set El with p(E1) = 1 such that, for any point t E Q, we have H(*,t)lE1= G(.,t)IE1. In particular, G turns out to be a measurable separable modification of
H. Proof. In view of Lemma 1, for each
Consequently, for any natural n
E
> 0, we can write
> 0, there exists a finite family of reals
belonging to Q and satisfying the conditions: (1) the length of each segment [t;,tytl] is less than l l n ; (2) if t and t' belong to some segment [t?, t7+"+], then
Moreover, we may choose the above-mentioned families
in such a way that the following conditions will be fulfilled, too: (3) for any n E N \ {O), the set Q, is contained in the set Qntl; (4) Q = U{Q, : n E N , n > 0).
Now, let us put
E' = n{dom(H(., t))
:
t E Q).
Obviously, we have p(E1) = 1. Further, for each natural n function G,: ~'x[O,l]-tR
> 0, define a
by the equalities
Gn(e,1) = H(e, 1). Evidently, the partial function Gn is measurable with respect to the product a-algebra of S and B([O, I]). Furthermore, the series
is convergent for any point t E [O,l]. Hence, for each t E [0, 11, we get
almost everywhere in E (with respect to p, of course). Let us put
for all those pairs (e, t ) E El x T for which the above-mentioned limsup exists. In this way, we obtain a partial mapping
The definition of G implies a t once that G is a measurable stochastic process stochastically equivalent to H and, for any point t E Q, we have
Let now t be an arbitrary point from [O,11 \ Q. Then there exists an increasing sequence
such that
NONDIFFERENTIABLE FUNCTIONS
In virtue of the definition of G, we easily obtain
for any point e E E 1 n d o m ( G ( . t, ) ) . This completes the proof of the lemma.
Remark 5. The process G of Lemma 2 is usually called a separable modification of the original process H. Note that the existence of a separable modification of a given process can be established in a much more general situation than in that described by Lemma 2. For our further purposes this lemma is completely sufficient. More deep results may be found in [I131 and [121]. It is interesting to mention here that the general theorem concerning the existence of a separable modification of a stochastic process essentially relies on the notion of a von Neumann topology (multiplicative lifting). For details, see e.g. [I211 where such an approach is developed. Lemma 3. Let and (/?n)nEN be two sequences of st~ictly positive real numbers, such that
and let { f n : n E N) be a sequence of random variables on ( E , S , p ) satasfying the relataons ~ ( { Ee E : Ifn(e)l > a n } ) < Pn Then there exists a p-measure zero set A
c
(n E N ) .
E such that, for any point
s's convergent.
Proof. For each n E N , let us denote
Then, according to our assumption,
Let us put
A = n n c ~ ( U r n Ern>nAm). ~,
CHAPTER
13
Then we obviously have p(A) = 0.
Take any point e from E \ A. There exists a natural number k for which
This means that, for each natural m
is fulfilled. Hence the series
> k, the inequality
CnEN Ifn(e)l is convergent.
Lemma 4. Let H be a stochastic process such that d o m ( H ) = E x [0, 11 and
+
for all t E [0, 11 and t r E [0, 11, where d > 0 is some fixed constant. Then there exists a stochastic process G satisfying the relations: 1) G and H are stochastically equivalent; 2) G is measurable; 8) G is separable with a set of separability
4)
for any point t E Q , we have
5) almost all (wath respect to p) trajectories o f G are continuous realvalued functions defined on the whole segment [0,1]. Proof. First of all, we may write
+
for any t E [O, 11 and t r E [0, 11. This immediately implies that H is stochastically continuous. Applying Lemma 2, we can find a process
satisfying relations 1) - 4). Indeed, relations 1) - 3) are satisfied in virtue of Lemma 2, and relation 4) is valid since d o m ( H ) = E x [O,l]. Let us denote
275
NONDIFFERENTIABLE FUNCTIONS
where k and m are assumed to be natural numbers. Obviously, random variable. Furthermore, we have
a,
is a
In view of Lemma 3, the series
is convergent almost everywhere in E , i.e, there exists a p-measure zero set A such that @m(e) < +m rn€N
C
for all elements e E E \ A. Now, we fix n E N and easily observe that if t E [O,l], t' E [O,1] and It - t'I < 2-", then, for some natural k, the number k/2" is less than or equal to 1 and
Evidently, IG(., t)
- G(*,t')l < (G(s,t) - G(., k/2")( + IG(., t') - G(., k/2")(.
But if, in addition, t E Q and t' E Q , then it can directly be checked that
which yields the relation
Utilizing the separability of G, we infer that there exists a p-measure zero set B having the following property: if e is an arbitrary element from E \ (A U B) and t and t' are any two points such that
then
But we know that, for e E E\(AUB), the series C, N cPrn(e)is convergent. Thus, we conclude that the trajectory G(e, .) is uniformly continuous. This immediately implies that G(e, .) is a restriction of a continuous real-valued function defined on 10, I]. So we may extend G to a new process in such a way that all trajectories of this process, corresponding to the elements from E \ ( A U B), turn out to be continuous on [0,1]. It can easily be seen that the new process (denoted by the same symbol G) is separable and measurable as well. Indeed, the separability of G holds trivially and the measurability of G follows from the fact that G is measurable with respect to e E E and is continuous with respect to t E [O, 11. Lemma 4 has thus been proved. We now are ready to establish the following result.
Theorem 3 . The W i e n e r measure pw induces a Borel probability naeasure p on the space C[O, 11, with propertaes analogous t o the corresponding properties of pw.
Proof. Indeed, we have the probability measure space
and the standard Wiener process W = ( p ~ . t ) ~ for ~ [ this ~ , ~space. ] In view of the preceding lemma, there exists a process G for the same space, such that: 1) W and G are stochastically equivalent; 2) G is measurable; 3) G is separable with a set of separability
4) for any point t E Q, we have W(.,t) = G(.,t); 5) almost all trajectories of G are continuous real-valued functions on [O, 11.
277
NONDIFFERENTIABLE FUNCTIONS
Let El denote the set of all those elements e E E = for which the trajectory G(e, .) is continuous on [O,l]. Obviously, pw(E1)= 1. Define a mapping (b : E1+CIO,l] ~
[
~
l
l
]
by the formula (e E E'). 4(e) = G(e, Observe that (b is measurable with respect to pw (this fact easily follows from the result of Exercise 6). So we can put s)
Since p is a homomorphic image of p,, we have P(X) = P W ( { E~ E : G(e, .) E X ) ) for each Borel subset X of C[O, 11. In particular, if a any two points of [0,1], then ~ ( { fE C[O, 11 : If(t)-f(t')l
= p, ({e E E
< a)) = pw({e E E
:
> 0 and t and t' are
IG(e,t)-G(e,tl))
< a))
1
: W(e, t) - W(e, tl)l < a)).
In a certain sense, we may identify p and p,. So it will be convenient to preserve the same notation pw for the obtained measure p. Irother words, we consider p, as a Borel probability measure on the space C[O, 11. At last, we are able to return to the question of the differentiability of continuous real-valued functions on [O,1] (from the point of view of p,). Namely, the following statement is true.
Theorem 4. Almost all (with respect to p,) finctions from C[O, 11 are nondafferentaable almost everywhere on [O, 11 (with respect to A). Proof. Let us introduce the set
D = {(f,t) E C[O, 11 x [O,1] : f is differentiable at t). It can easily be checked that the set D is (p, x A)-measurable in the product space C[O, 11 x [ O , l ] . So, taking into account the F'ubini theorem, it suffices to show that, for each t E [O, I], the set
is of pw-measure zero. In order to do this, we first observe that the inclusion
CHAPTER 13
278
is satisfied. Hence, it suffices to prove, for each n E N , that
where Dt,n
= {f E C[O,lI :
/imsupl,l,~+
If
(t
+ r ) - f(t)J/lrl < n ) .
Further, one can easily verify that
and Dt,n,a,r =
{f E C[O, 11
:
If (t + r) - f (t)I/IrI < n).
Thus, it remains to demonstrate that But, for any r satisfying 0 < lrI < 6 , we may write
(&r)''/'
/nli'l/'
e ~ p ( - ~ ~ / 2=) 0d (~1 ~ 1 ~ / ~ ) .
-nlrll/a
This immediately implies the desired result, since lrI arbitrarily small.
> 0 can be chosen
Remark 6. A more general result obtained by Wiener and LBvy holds true; namely, they proved that almost all (with respect to p w ) functions from C[O, 11 are nowhere differentiable on [O,l]. Briefly speaking, almost all trajectories of the modificated Wiener process are nowhere' differentiable on [ O , l ] . For extensive information concerning the relationships between stochastic processes and Brownian motion, we refer the reader to the fundamental monograph by Levy [96]. Remark 7. As mentioned earlier, the standard Wiener process is a very particular case of a Gaussian process. Gaussian processes form a natural class of stochastic processes which have many interesting properties (see, e.g., [36], [113], [121], [146], [23]) and are important from the point of view of numerous applications.
Bibliography
[I] A. Andretta, A. Marcone. Ordinary differential equations and descriptive set theory: uniqueness and globality of solutions of Cauchy problems in one dimension. Fund. Math., vol. 153, no. 2, pp. 157 190, 1997. [2] M. Balcerzak. Another nonmeasurable set with property (so). Real Analysis Exchange, vol. 17, no. 2, pp. 781 - 784, 1991 - 1992. [3] M. Balcerzak. Some remarks on sup-measurability. Real Analysis Exchange, vol. 17, no. 2, pp. 597 - 607, 1991 - 1992. [4] M. Balcerzak. Typical properties of continuous functions via the Vietoris topology. Real Analysis Exchange, vol. 18, no. 2, pp. 532 536, 1992 - 1993. [5] S. Baldwin. Martin's axiom implies a stronger version of Blumberg's theorem. Real Analysis Exchange, vol. 16, pp. 67 - 73, 1990 - 1991. r Baire'sche Kategorie gewisser Funktionenmen[6] S. Banach: ~ b e die gen. Studia Math., vol. 3, pp. 174 - 179, 1931. [7] S. Banach, C. Kuratowski. Sur une gdndralisation du probldme de la mesure. Fund. Math., vol. 14, pp. 127 - 131, 1929. [8] S. Banach, H. Steinhaus. Sur le principe de la condensation de singularitds. Fund. Math., vol. 9, pp. 50 - 61, 1927. [9] F. Bernstein. Zur Theorie der trigonometrischen Reihen, Sitzungsber. Sachs. Akad. Wiss. Leipzig. Math.-Natur. K1. 60, pp. 325 - 338, 1908. [lo] J . Blazek, E. Bordk, J. Malj. On Kopcke and Pompeiu functions. pro pest. mat., vol. 103, pp. 53 - 61, 1978.
as.
280
B I B LI OG R A PHY
[ l l ] H. Blumberg. New properties of all real functions. Trans. Amer. Math. Soc., vol. 24, pp. 113 - 128, 1922. [12] V.G. Boltjanski. The Hilbert Third Problem. Moscow: Izd. Nauka, 1977 (in Russian). [13] N. Bourbaki. Set Theory. Moscow: Izd. Mir, 1965 (in Russian, translation from French). [14] N. Bourbaki. Integration. Moscow: Izd. Mir, 1977 (in Russian; translation from French). [15] J .B. Brown. Restriction theorems in real analysis. Real Analysis Exchange, vol. 20, no. 2, pp. 510 - 526, 1994 - 1995. [16] J.B. Brown. Variations on Blumberg's theorem. Real Analysis Exchange, vol. 9, pp. 123 - 137, 1983 - 1984. [17] J.B. Brown, G.V. Cox. Classical theory of totally imperfect spaces. Real Analysis Exchange, vol. 7, pp. 1 - 39, 1982. [I81 A. Bruckner. Differentiation of Real Functions. Berlin: SpringerVerlag, 1978. [19) A. Bruckner, J . Haussermann. Strong porosity features of typical continuous functions. Acta Math. Hung., vol. 45, no. 1 - 2, pp. 7 13, 1985. [20] A. Bruckner, J .L. Leonard. Derivatives. Amer. Math. Monthly, vol. 73, no. 4, part 11, pp. 24 - 56, 1966. [21] A.M. Bruckner, R.J. O'Malley, B.S. Thomson. Path derivatives: a unified view of certain generalized derivatives. Trans. Amer. Math. Soc., vol. 283, no. 1, pp. 97 - 125, 1984. [22] L. Bukovskjr, N.N. Kholshchevnikova, M. Repickjr. Thin sets of harmonic analysis and infinite combinatorics. Real Analysis Exchange, vol. 20, no. 2, pp. 454 - 509, 1994 - 1995. [23] V.V. Buldygin, A.B. Kharazishvili. Brunn-Minkowski Inequality and its Applications. Kiev: Izd. Naukova Dumka, 1985 (in Russian). [24] J . Cichori, A. Kharazishvili, B. Wqglorz. On selectors associated with some subgroups of the real line. Bull. Acad. Sci. of Georgian SSR, vol. 144, no. 2, 1991.
281
BI B LI OG R A PHY
[25] J . Cichori, A. Kharazishvili, B. Weglorz. On sets of Vitali's type. Proc. Amer. Math. Soc., vol. 118, no. 4, 1993. [26] J. Cichori, A. Kharazishvili, B. Weglorz. Subsets of the Real Line. Lbdi: Lbdi University Press, 1995. 1271 J. Cichori, M. Morayne. On differentiability of Peano type functions, 111. Proc. Amer. Math. Soc., vol. 92, no. 3, pp. 432 - 438, 1984. 1281 K. Ciesielski. Set-theoretic real analysis. Journal of Applied Analysis, vol. 3, no. 2, pp. 143 - 190, 1997. [29] K. Ciesielski, L. Larson, K. Ostaszewski. Differentiability and density continuity. Real Analysis Exchange, vol. 15, no. 1, pp. 239 - 247, 1989 - 1990. [30] K. Ciesielski, L. Larson, K. Ostaszewski. I-density continuous functions. Memoirs of the Amer. Math. Soc., vol. 107, no. 515, pp. 1 132, 1994. [31] P.M. Cohn. Universal Algebra. London: Harper and Row, 1965. 1321 P. Corazza. Ramsey Sets, the Ramsey Ideal and Other Classes over R. preprint . [33] M. Dehn. Uber den Rauminhalt. Math. Ann., vol. 55, pp. 465 - 478, 1902. [34] C. Dellacherie. Capacitds et Springer-Verlag, 1972.
Processus Stochastiques.
Berlin:
[35] A. Denjoy. Sur les fonctions dkrivkes sommables, Bull. Soc. Math. France, vol. 43, pp. 161 - 248, 1915. [36] J.L. Doob. Stochastic Processes. New York: Wiley and Sons, Inc., 1953. [37] D. Egoroff. Sur les suites de fonctions mesurables. C. R. Acad. Sci. Paris, vol. CLII, 1911. 1381 I. Ekeland. Elements d'Economie Mathematique. Paris: Hermann, "979. [39] P. Erdos, S. Kakutani. On nondenumerable graphs. Bull. Amer. Math. Soc., vol. 49, pp. 457 - 461, 1943.
282
B I B L I OG R A PHY
[40] P. Erdos, R.D. Mauldin. The nonexistence of certain invariant measures. Proc. Amer. Math. Soc., vol. 59, pp. 321 - 322, 1976. [41] P. Erdos, A.H. Stone. On the sum of two Bore1 sets. Proc. Amer. Math, Soc., vol. 25, no. 2, pp. 304 - 306, 1970. [42] M.P. Ershov. Measure extensions and stochastic equations. Probability Theory and its Applications, vol. 19, no. 3, 1974 (in Russian). [43] M. Evans. On continuous functions and the approximate symmetric derivatives. Colloq. Math., vol. 31, pp. 129 - 136, 1974. [44] G. Fichtenholz. Sur une fonction de deux variables sans intkgrale double. Fund. Math., vol. 6, pp. 30 - 36, 1924. [45] C. Freiling. A converse to a theorem of Sierpiriski on almost symmetric sets. Real Analysis Exchange, vol. 15, no. 2, pp. 760 - 767, 1989 - 1990. [46] C. Freiling. Axioms of symmetry: throwing darts at the real number line. The Journal of Symbolic Logic, vol. 51, pp. 190 - 200, 1986. [47l D.H. Fremlin. Measure additive coverings and measurable selectors. Dissertationes Mathematicae, vol. CCLX, 1987. [48] H. Friedman. A consistent Fubini-Tonelli theorem for non-measurable functions. Illinois J . Math., vol. 24, pp. 390 - 395, 1980. [49] B.R. Gelbaum, J.M.H. Olmsted. Counterexamples in Analysis. San Francisco: Holden-Day, 1964. [50] C. Goffman. Everywhere differentiable functions and the density topology. Proc. Amer. Math. Soc., vol. 51, pp. 250 - 251, 1975. [51] C. Goffman, C.J. Neugebauer, T. Nishiura. Density topology and approximate continuity. Duke Mathem. Journal, vol. 28, pp. 497 505, 1961. [52] Z. Grande, J . Lipiriski. Un exemple d'une fonction sup-mesurable qui n'est pas mesurable. Colloq. Math., vol. 39, pp. 77 - 79, 1978. [53] P.R. Halmos. Measure Theory. New York: D.Van Nostrand, 1950. 9
[54] G. amel el. Eine Basis aller Zahlen und die unstetigen Losungen der Funktionalgleichung: f (x y) = f ( x ) + f (y). Math. Ann., vol. 60, pp. 459 - 462, 1905.
+
B I BL I OG R A PHY
283
[55] Handbook of Mathematical Logic (edited by J . Barwise). Amsterdam: North-Holland Publishing Comp., 1977. [56] Handbook of Mathematical Logic, Part 2. Moscow: Izd. Nauka, 1982 (in Russian, translation from English). [57] E. Hewitt, K. Ross. Abstract Harmonic Analysis, vol. 1. Berlin: Springer-Verlag, 1963. [58] E.W. Hobson. Theory of Functions of a Real Variable, 11. New York: Dover, 1957. [59] D.H. Hyers, S.M. Ulam. Approximately convex functions. Proc. Amer. Math. Soc., vol. 3, pp. 821 - 828, 1952. [60] V. ~ a r n i k Sur . la dkrivabilitk des fonctions continues. Spisy Privodov, Fak. Univ. Karlovy, vol. 129, pp. 3 - 9, 1934. [61] J . Jasinski, I. Reclaw. Restrictions to continuous and pointwise discontinuous functions. Real Analysis Exchange, vol. 23, no. 1, pp. 161 - 174, 1997 - 1998. [62] B. Jessen. The algebra of polyhedra and the Dehn-Sydler theorem. Math. Scand., vol. 22, pp. 241 - 256, 1968. [63] S. Kaczmarz, H. Steinhaus. Theorie der hen. Warszawa-Lwow: Monogr. Matem., 1935.
Orthogonalrei-
[64] Y. Katznelson, K. Stromberg. Everywhere differentiable, nowhere monotone functions. Amer. Math. Monthly, vol. 81, pp. 349 - 354, 1974. [65] A.S. Kechris. Classical Descriptive Set Theory. New York: SpringerVerlag, 1995. [66] J.L. Kelley. General Topology. New York: D. Van Nostrand, 1955. [67] A.B. Kharazishvili. Applications of Set Theory. Tbilisi: Izd. Tbil. Gos. Univ., 1989 (in Russian). 1681 A.B. Kharazishvili. Certain types of invariant measures. Dokl. Akad. Nauk SSSR, vol. 222, no. 3, pp. 538 - 540, 1975 (in Russian). [69] A.B. Kharazishvili. Invariant Extensions of Lebesgue Measure. Tbilisi: Izd. Tbil. Gos. Univ., 1983 (in Russian).
284
BIBLI OG R A P H Y
[70] A.B. Kharazishvili. Martin's axiom and I?-selectors. Bull. Acad. Sci. of Georgian SSR, vol. 137, no. 2, 1990 (in Russian). [71] A.B. Kharazishvili. On sections of analytic sets. Bull. Acad. Sci. of Georgia, vol. 147, no. 2, pp. 223 - 226, 1993 (in Russian). [72] A.B. Kharazishvili. On translations of sets and functions. Journal of Applied Analysis, vol. 1, no. 2, pp. 145 - 158, 1995. [73] A.B. Kharazishvili. Selected Topics of Point Set Theory. L6di: Mdf University Press, 1996. [74] A.B. Kharazishvili. Some applications of Hamel bases. Bull. Acad. Sci. Georgian SSR, vol. 85, no. 1, pp. 17 - 20, 1977 (in Russian). [75] A. Kharazishvili. Some questions concerning invariant extensions of Lebesgue measure. Real Analysis Exchange, vol. 20, no. 2, pp. 580 - 592, 1994 - 1995. [76] A.B. Kharazishvili. Some Questions of Set Theory and Measure Theory. Tbilisi: Izd. Tbil. Gos. Univ., 1978 (in Russian). [77] A.B. Kharazishvili. Some questions of the theory of invariant measures. Bull. Acad. Sci. of GSSR, vol. 100, no. 3, 1980 (in Russian). [78] A.B. Kharazishvili. Some remarks on density points and the uniqueness property for invariant extensions of the Lebesgue measure. Acta Universitatis Carolinae - Mathematica et Physica, vol. 35, no. 2, pp. 33 - 39, 1994. [79] A. Kharazishvili. Some remarks on the property (N) of Luzin. Annales Mathematicae Silesianae, vol. 9, pp. 33 - 42, 1995. [80] A.B. Kharazishvili. Vitali's Theorem and Its Generalizations. Tbilisi: Izd. Tbil. Gos. Univ. 1991 (in Russian). [81] B. King. Some remarks on difference sets of Bernstein sets. Real Analysis Exchange, vol. 19, no. 2, pp. 478 - 490, 1993 - 1994. [82] E.M. Kleinberg. Infinitary Combinatorics and the Axiom of Determinateness. Berlin: Springer-Verlag, 1977. [83] A.N: Kolmogoroff. Sur la possibilitd de la definition gkndrale de la deride, de l'integrale et de la sommation des series divergentes. C. R. Acad. Sci. Paris, vol. 180, pp. 362 - 364, 1925.
B I BLI OG RA P HY
285
[84] P. KornjSth. A note on set mappings with meager images. Studia Scientiarum Mathematicarum Hungarica, vol. 30, pp. 461 - 467, 1995. 1851 M.A. Krasnoselski, A.V. Pokrovski. Systems with Hysteresis. Berlin: Springer-Verlag, 1988. [86] M. Kuczma. An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality. Katowice: PWN, 1985. [87] K.Kunen. Random and Cohen reals. In: Handbook of Set-Theoretic Topology, edited by K.Kunen and J.E.Vaughan, Amsterdam: NorthHolland, 1984. [88] K. Kunen. Set Theory. Amsterdam: North-Holland, 1980. [89] K. Kuratowski. Topology, vol. 1. New York: Academic Press, 1966. [go] K. Kuratowski. Topology, vol. 2. New York: Academic Press, 1968. [91] K. Kuratowski, A. Mostowski. Set Theory. Amsterdam: NorthHolland Publishing Comp., 1967. [92] K. Kuratowski, C. Ryll-Nardzewski. A general theorem on selectors. Bull. Acad. Polon. Sci., Ser. Math., vol. 13, no. 6, pp. 397 - 402, 1965. [93] M. Laczkovich, G. Petruska. On the transformers of derivatives. Fund. Math., vol. 100, pp. 179 - 199, 1978. [94] M.A. Lavrentieff. Contribution S la thkorie des bles homiomorphes. Fund. Math., vol. 6, 1924.
ensem-
[95] M.A. Lavrentieff. Sur une iquation differentielle du premier ordre. Math. Zeitschrift, vol. 23, pp. 197 - 209, 1925. [96] P. Lkvy. Processus Stochastiques et Mouvement Brownien. Paris: Gauthier-Villars, 1965. [97] N. Lusin. Lesons sur les Ensembles Analytiques et leurs Applications. Paris: Gauthier - Villars, 1930. [98] N. Lusin. Sur une problkme de M. Baire. C.R. Acad. Sci. Paris, vol. 158, p. 1259, 1914.
286
BIBLI O G R A P HY
1991 N.N.Luzin. Integral and Trigonometric Series. Moscow: Izd. GITTL, 1956 (in Russian). [loo] D. Maharam. On a theorem of von Neumann. Proc. Amer. Math. SOC.,vol. 9, pp. 987 - 994, 1958. [loll J . Malf. The Peano curve and the density topology. Real Analysis Exchange, vol. 5, pp. 326 - 329, 1979 - 1980. [I021 E. Marczewski, C. Ryll-Nardzewski. Sur la mesurabilitk des fonctions de plusieurs variables. Annales de la Societk Polonaise de Mathkmatique, vol. 25, pp. 145 - 155, 1952. [I031 R.D. Mauldin. The set of continuous nowhere differentiable functions. Pacific J . Math., vol. 83, no. 1, pp. 199 - 205, 1979; vol. 121, no. 1, pp. 119 - 120, 1986. [I041 S. Mazurkiewicz. Sur les fonctions non-derivables. Studia Math., vol. 3, pp. 92 - 94, 1931. [I051 E. Michael. Continuous selections, I. Ann. of Math., vol. 63, no. 2, pp. 361 - 382, 1956. [I061 E. Michael. Selected selection theorems. Amer. Math. Monthly, vol. 63, pp. 233 - 238, 1956. [I071 A.W. Miller. Special subsets of the real line. In: Handbook of SetTheoretic Topology, edited by K.Kunen and J.Vaughan, Amsterdam: North-Holland, 1984. [I081 M. Morayne. On differentiability of Peano type functions. Coll. Math., vol. 53, no. 1, pp. 129 - 132, 1987. [log] M. Morayne. On differentiability of Peano type functions, 11. Coll. Math., vol. 53, no. 1, pp. 133 - 135, 1987. [I101 J.C. Morgan 11. Point Set Theory. New York: Marcel Dekker, Inc., 1990. [ I l l ] I.P. Natanson. The Theory of Functions of a Real Variable. Moscow: Izd. Nauka, 1957 (in Russian). [I121 T . Natkaniec. Almost Continuity. Bydgoszcz: WSP, 1992. [I131 J . Neveu. Bases Mathematiques du Calcul des Probabilitks. Paris: Masson et Cie, 1964.
BIBLIOGRAPHY
287
[I141 W. Orlicz. Collected Papers, vols. 1 and 2. Warszawa: PWN, 1988. [I 151 W. Orlicz. Zur Theorie der Differentialgleichung y' = f (x, y). Polska Akademia Umiejetnosci, Krakow, pp. 221 - 228, 1932. [I161 J.C. Oxtoby. Cartesian products of Baire spaces. Fund. Math., vol. XLIX, pp. 157 - 166, 1961. [I171 J.C. Oxtoby. Measure and Category. Berlin: Springer-Verlag, 1971. [I181 J. Pawlikowski. The Hahn-Banach theorem implies the BanachTarski paradox. Fund. Math., vol. 138, pp. 21 - 22, 1991. [I191 I.G.Petrovski. Lectures on the Theory of Ordinary Differential Equations. Moscow: GITTL, 1949 (in Russian). [I201 J . Raisonnier. A mathematical prof of S.Shelah's theorem on the measure problem and related results. Israel Journal of Mathematics, vol. 48, no. 1, 1984. [I211 M.M. Rao. Stochastic Processes and Integration. Sijthoff and Noordhoff, Alphen aan den R i n , 1979. 11221 D. Repovs, P.V. Semenov. E.Michael's theory of continuous selections; development and applications. Uspekhi Mat. Nauk, vol. 49, no. 6 (300), 1994 (in Russian). [I231 C.A. Rogers. A linear Borel set whose difference set is not a Borel set. Bull. London Math. Soc., vol. 2, pp. 41 - 42, 1970. [I241 F. Rothberger. Eine ~ ~ u i v a l e nzwischen z der Kontinuumhypothese und der Existenz der Lusinschen und Sierpiriskischen Mengen. Fund. Math., vol. 30, pp. 215 - 217, 1938. [I251 S. Ruziewicz. Sur une proprietk de la base hamelienne. Fund. Math., vol. 26, pp. 56 - 58, 1936. [I261 C. Ryll-Nardzewski, R. Telgarski. The nonexistence of universal invariant measures. Proc. Amer. Math. Soc., vol. 69, pp. 240 - 242, 1978. [I271 S. Saks. Theory of the Integral. Warszaw&Lw6w, 1937. [I281 E.A. Selivanovski. On a class of effective sets. Mat. Sbor., vol. 35, pp. 379 - 413, 1928 (in Russian).
288
BIBLIOGRAPHY
[I291 S. Shelah. Can you take Solovay's inaccessible away? Israel 3 . Math., vol. 48, no. 1, pp. 1 - 47, 1984. [I301 J . Shinoda. Some consequences of Martin's axiom and the negation of the continuum hypothesis. Nagoya Math. J., vol. 49, pp. 117 .125, 1973. [I311 J . Shipman. Cardinal conditions for strong Fubini theorems. Trans. Amer. Math. Soc., vol. 321, pp. 465 - 481, 1990. [I321 J .R. Shoenfield. Measurable cardinals. In: Logic Colloquium'69, Amsterdam: North-Holland, 1971. [I331 J.W. Shragin. Conditions for measurability of superpositions. Dokl. Akad. N.auk SSSR, vol. 197, pp. 295 - 298, 1971, (in Russian). [I341 W. Sierpiriski. Cardinal and Ordinal Numbers. Warszawa: PWN, 1958. [I351 W. Sierpiriski. Hypothkse du Continu. New York: Chelsea Publ. Co., 1956. [I361 W. Sierpiriski, Les correspondances multivoques et I'axiome du choix. Fund. Math., vol. 34, pp. 39 - 44, 1947. [I371 W. Sierpiriski. Oeuvres Choisies, Volumes I - 111. Warszawa: PWN, 1972 - 1976. 11381 W. Sierpiliski. Sur la question de la mesurabiliti de la base de M.Hame1. Fund. Math., vol. 1, pp. 105 - 111, 1920. [I391 W. Sierpiriski. Sur l'ensemble de distances entre les points d'un ensemble. Fund. Math., vol. 7, pp. 144 - 148, 1925. [I401 W. Sierpiriski. Sur le produit combinatoire de deux ensembles jouissant de la proprikti C. Fund. Math., vol. 24, pp. 48 - 50, 1935. [I411 W. Sierpiriski. Sur une fonction non mesurable partout presque symetrique. Acta Litt. Scient., Szeged, vol. 8, pp. 1 - 6, 1936. [I421 W. Sierpiliski. Sur un thiorime kquivalent B l'hypothise du continu. Bull. Internat. Acad. Sci. Cracovie Ser. A, pp. 1 - 3, 1919. [I431 W. Sierpiriski, A. Zygmund. Sur une fonction qui est discontinue sur tout ensemble de puissance du continu. Fund. Math., vol. 4, pp. 316 - 318, 1923.
289
BIBLIOGRAPHY
[I441 J.C. Simrns. Sierpiriski's theorem. Simon Stevin, vol. 65, no. 1 - 2, pp. 69 - 163, 1991. [I451 S.G. Simpson. Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations? Journal of Symbolic Logic, vol. 49, pp. 783 - 802, 1984. [I461 A.V. Skorokhod. Integration in Hilbert Space. Moscow: Izd. Nauka, 1975 (in Russian). [I471 B.S. Sodnomov. An example of two G6-sets whose arithmetical sum is not Bore1 measurable. Dokl. Akad. Nauk SSSR, vol. 99, pp. 507 510, 1954 (in Russian). [I481 B.S. Sodnomov. On arithmetical sums of sets. Dokl. Akad. Nauk SSSR, vol. 80, no. 2, pp. 173 - 175, 1951 (in Russian). [I491 S. Solecki. Measurability properties of sets of Vitali's type. Proc. Amer. Math. Soc., vol. 119, no. 3, pp. 897 - 902, 1993. [I501 S. Solecki. On sets nonmeasurable with respect to invariant measures. Proc. Amer. Math. Soc., vol. 119, no. 1, pp. 115 - 124, 1993. 11511 R.M. Solovay. A model of set theory in which every set of reals is Lebesgue measurable. Ann. Math., vol. 92, pp. 1 - 56, 1970. [I521 R.M. Solovay. Real-valued measurable cardinals. Proceedings of Symposia in Pure Mathematics, vol. XII, Axiomatic set theory, Part 1, Providence 1971, pp. 397 - 428. [I531 J . Stallings. Fixed point theorems for connectivity maps. Fund. Math., vol. 47, pp. 249 263, 1959.
-
11541 H.Steinhaus. Sur les distances des points de mesure positive. Fund. Math., vol. 1, pp. 93 - 104, 1920. [I551 J.-P. Sydler. Conditions nkcessaires et suffisantes pour 1'6quivalence des polykdres de l'espace euclidien B trois dimensions. Comment. Math., Helv., vol. 40, pp. 43 - 80, 1965. [I561 E. Szpilrajn (E. Marczewski). Sur une classe de fonctions de M. Sierpiriski et la classe correspondante d'ensembles. Fund. Math., vol. 24, pp. 17 - 34, 1935.
290
BIBLIOGRAPHY
[I571 E. Szpilrajn (E. Marczewski). The characteristic function of a sequence of sets and some of its applications. Fund. Math., vol. 31, pp. 207 - 223, 1938. [I581 F. Tall. The density topology. Pacific Journal of Mathematics, vol. 62, pp. 275 - 284, 1976. [I591 T . Traynor. An elementary proof of the lifting theorem. Pacific J Math., vol. 53, pp. 267 - 272, 1974. [I601 S. Ulam. A Collection of Mathematical Problems. New York: Interscience, 1960. [I611 S. Ulam. Zur Masstheorie in der allgemeinen Mengenlehre. Fund. Math., vol. 16, pp. 140 - 150, 1930. [I621 G. Vitali. Sul problema della misura dei gruppi di punti di una retta. Nota, Bologna, 1905. [I631 S. Wagon. The Banach-Tarski Paradox. Cambridge: Cambridge University Press, 1985. [I641 J.T. Walsh. Marczewski sets, measure and the Baire property 11. Proc. Amer. Math. Soc., vol. 106, no. 4, pp. 1027 - 1030, 1989. [I651 C. Weil. On nowhere monotone functions. Proc. Amer. Math. Soc., vol. 56, pp. 388 - 389, 1976. [I661 W. Wilczyriski. A category analogue of the density topology, approximate continuity and the approximate derivative. Real Analysis Exchange, vol. 10, no. 2, pp. 241 - 265, 1984 - 1985. [I671 W. Wilczyriski, A. Kharazishvili. On translations of measurable sets and of sets with the Baire property. Bull. Acad. Sci, of Georgia, vol. 145, no. 1, pp. 43 - 46, 1992 (in Russian). [I681 Z. Zahorski. Sur la premiere derivke. Trans. Amer. Math. Soc., vol. 69, pp. 1 - 54, 1950.
Index
absolutely continuous function, 62 absolutely nonmeasurable function, 173 admissible functional, 182 almost symmetric function, 155 analytic set, 16 analytic space, 16 antichain, 7 (A)-operation, 16 approximate derivative, 29 approximately continuous function, 84 approximately differentiable function, 85 Axiom of Choice, 1 Axiom of Dependent Choices, 6 Axiom of Projective Determinacy, 257 axioms of set theory, 1
Baire property, 5 Baire property in the restricted sense, 20 Banach-Kuratowski matrix, 203 Banach condition, 205 Banach-Tarski paradox, 136 Bernstein set, 119 bijection, 12 binary relation, 5 Blumberg space, 100 Borel isomorphism, 17 Borel mapping, 17 Borel measure, 16 Borel set, 16
INDEX
Bore1 a-algebra, 16
Cantor discontinuurn, 38 canonical Baire space, 47 Cantor space, 38 Cantor type function, 34 Caratht5odory conditions, 220 cardinality of the continuum, 4 cardinal number, 3 Cartesian product, 3 Cauchy functional equation, 137 Cauchy problem, 237 chain, 7 class of sets, 2 closed graph, 20 C-measurable mapping, 232 compact topological space, 15 continuous mapping, 17 Continuum Hypothesis, 4 convex function, 149 countable chain condition, 8 countable form of the Axiom of Choice, 15 countable set, 3 C-set, 232 cylindrical a-algebra, 262
Darboux property, 57 decreasing function, 55 density point, 29 density topology, 19 derived number, 58 difference of sets, 2 dihedral angle, 158 disjoint family of sets, 13 domain of a partial function, 11 duality between two vector spaces, 51
INDEX
element of a set, 1 empty set, 2 equivalence relation, 6 Euclidean space, 5 extension of a partial function, 11
family of sets, 2 finite set, 3 first category set, 19 function, 10 functional graph, 10
Gaussian distribution, 261 generalized Cantor discontinuum, 43 Generalized Continuum Hypothesis, 4 generalized derivative, 29 generalized integral, 29 generalized Luzin set, 171 generalized Sierpiriski set, 171
Hamel basis, 5 Hausdorff metric, 46 Hilbert cube, 16 Hilbert dimension, 179 Hilbert space, 179
increasing function, 55 infinite set, 3 initial condition, 237 injection, 11 intersection of sets, 2 iterated integrals, 199
INDEX
Jensen inequality, 149
largest element, 9 Lebesgue measure, 5 limit ordinal, 3 linearly ordered set, 9 Lipschitz condition, 77 local maximum, 77 local minimum, 77 lower semicontinuous function, 21 lower semicontinuous set-valued mapping, 21 Luzin set, 19
mapping, 10 Marczewski set, 120 Martin's Axiom, 8 mathematical expectation, 260 maximal element, 9 measurable stochastic process, 266 membership relation, 1 minimal element, 9
normal distribution, 261 normal topological space, 101 nowhere dense set, 19 nowhere approximately differentiable function, 83 nowhere differentiable function, 27
one-to-one correspondence, 12 ordered pair, 2 ordinal number, 3 ordinary differential equation, 217 oscillation of a function, 106
INDEX
partial function, 10 partial mapping, 10 partial order, 7 partially ordered set, 7 partition associated with an equivalence relation, 7 Peano type function, 38 perfect set, 27 Polish space, 16 polyhedron, 157 principle of condensation of singularities, 253 probability distribution, 258 probability space, 258 projective set, 18
quasicompact topological space, 14
Radon measure, 17 Radon space, 17 random variable, 258 range of a partial function, 11 real line, 4 regular cardinal, 3 regular ordinal, 3 relation, 1 restriction of a partial function, 11
selector, 14 semicompact family of functions, 187 separable measure, 259 separable stochastic process, 270 set, 1 set-valued mapping, 12 Sierpidski-Erdos Duality Principle, 45
INDEX
Sierpiriski set, 159 Sierpiriski's partition of the plane, 197 Sierpiriski-Zygmund function, 95 simple discontinuity point, 56 singular cardinal, 3 singular monotone function, 55 singular ordinal, 3 smallest element, 9 Steinhaus property, 114 stochastic process, 266 stochastically continuous process, 266 stochastically equivalent processes, 266 strictly decreasing function, 55 strictly increasing function, 55 strongly measure zero set, 166 subset, 1 sup-continuous mapping, 217 sup-measurable mapping, 217 surjection, 12 Suslin space, 16 symmetric derivative, 28 symmetric difference of sets, 2 symmetric group, 12
topological group, 24 topological structure, 15 topological weight of a space, 23 totally imperfect set, 119 transformation of a set, 12
Ulam transfinite matrix, 142 uncountable set, 3 union of sets, 2 universal measure zero space, 164 universal object, 54 upper $emicontinuous function, 21
INDEX
vector space, 5 Vitali covering, 29 Vitali partition, 118 Vitali set, 118 von Neumann topology, 93
weakly sup-measurable mapping, 217 well ordered set, 9 Wiener measure, 258 Wilczyriski's topology, 94
Zermelo-Fkaenkel set theory, 1 Zorn Lemma, 9
