Book Review

Topology Atlas Document # topd-02 | Production Editor: Thomas M. Zachariah

Book Review

Krzysztof Ciesielski

Book Review From Volume 5, of TopCom

Strange Functions in Real Analysis
by A. B. Kharazishvili
Marcel Dekker, Inc., New York and Basel, 2000, ISBN 0-8247-0320-0

As the title suggests, the subject of this book is concentrated around different strange (or singular) objects, mainly functions, which appear in real analysis in different contexts. This includes such classical, but counterintuitive, examples as the Peano curve, i.e., a continuous function from [0, 1] onto [0, 1]² (chapter 1), a strictly increasing function from R to R whose derivative is equal to zero almost everywhere (chapter 2), an everywhere differentiable nowhere monotone function f: R --> R (chapter 3), or a continuous nowhere (approximately) differentiable function (chapter 4). The constructions of the examples mentioned above are described in the text in detail. The author also stresses that these constructions are quite simple from the set-theoretical point of view - they do not require the use of the Axiom of Choice beyond, possibly, the Dependent Choice Axiom. This relative set-theoretical simplicity changes after chapter 4, when the Axiom of Choice and transfinite induction start playing a crucial role. These techniques are used, in particular, in the constructions of non-measurable sets (so also functions): the Vitali set and the Bernstein set (chapter 6). Transfinite induction is also used in chapter 5 for the construction of a Sierpinski-Zygmund function, that is, a function f: R --> R such that its restriction f|_X is discontinuous for any subset X of R of the cardinality of the continuum. This example is used to setup the limitation for possible generalizations of a Blumberg theorem (for an arbitrary function from R to R there exists a dense subset D of R such that f|_D is continuous), the proof of which is also included in the text. In chapter 7 the author discusses the notion of Hamel basis and describes discontinuous additive functions, that is functions f: R --> R which satisfy the Cauchy functional equation: f(x + y) = f(x) + f(y). From chapter 8 on the author starts using yet another set theoretical tool - the Continuum Hypothesis CH. Thus in chapter 8 he shows that CH implies the existence of Luzin and Sierpinski's sets and discusses some of their applications to real functions. The main result presented in chapter 10 is the equivalence of CH and the decomposition of the plane into two sets A and B such that A (respectively, B) has all vertical (respectively, horizontal) sections countable. This decomposition is also used to describe a Sierpinski's example of a function f: [0, 1]² --> [0, 1] for which the two iterated integrals of f (w.r.t. the first coordinate and then the second coordinate; w.r.t the second coordinate and then the fieat coordinate) exist but are not equal. Finally, in chapter 11 the author proves that CH implies the existence of a non-measurable function which is sup-measurable, while in chapter 12 he shows that sup-measurable functions have very interesting consequences in the theory of ordinary differential equations.

Assessment of the book is a bit difficult. On the one hand, the choice of examples presented in the text seems to be a very good one. It consists of examples which are either the most classical (as described in chapters 1-4) or those around which some recent research progress has been made. Although in many cases the new results have not been proved in the text (they require essentially more sophisticated set theoretical methods than those used in the book) they are stated and used in the discussion of the material presented. Also the order in which the results are presented seems to be very well chosen. All of this makes the book a very interesting text to read. On the other hand, the presentation of the material is rather unusual. Although it is certainly well accessible for real analysis researchers, it is unclear to what audience the text is aimed. It seems that the focus on the audience changed with progress on writing the book. Thus, the first four chapters seem to be aimed at an audience that has just been through a course on Lebesgue measure and integration. In this part most of the proofs are written in full detail and many of the results presented are standard facts from real analysis (e.g., theorems 1 and 2 from chapter 2) or topology (e.g., theorem 2 from chapter 1). At the same time, however, some exercises (like ex. 12, page 42) seem to require essentially more knowledge. With progress on the book the details of the proofs presented slowly disappear, and the proofs are replaced by sequences of exercises and remarks which are presented in an order that outlines a proof. (See, e.g., chapter 10.) Although this kind of presentation is usually sufficient for experts to follow the arguments, usually it is not appropriate for an audience with less expertise. It should be also mentioned that the chapters in the second part of the text are quite independent of each other, with many facts and definitions repeated in each of them independently. This makes the second part of the book more a collection of short survey articles which can be read quite independently of each other.