Strange Functions in Real Analysis

441 Pages·2017·2.229 MB·English

by  Alexander Kharazishvili

“K29544” — 2017/8/12
STRANGE
FUNCTIONS IN
REAL ANALYSIS
THIRD EDITION
“K29544” — 2017/8/12
“K29544” — 2017/8/12
STRANGE
FUNCTIONS IN
REAL ANALYSIS
THIRD EDITION
Alexander Kharazishvili
Razmadze Mathematical Institute
Tbilisi, Georgia
“K29544” — 2017/8/12
CRC Press
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“K29544” — 2017/8/24
Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Chapter 0. Introduction: Basic concepts . . . . . . . . . . 1
Chapter 1. Cantor and Peano type functions . . . . . . . 33
Chapter 2. Functions of first Baire class . . . . . . . . . . 53
Chapter 3. Semicontinuous functions
that are not countably continuous . . . . . . . . . . . . . . 71
Chapter 4. Singular monotone functions . . . . . . . . . . 81
Chapter 5. A characterization of constant
functions via Dini’s derived numbers . . . . . . . . . . . . 95
Chapter 6. Everywhere differentiable
nowhere monotone functions . . . . . . . . . . . . . . . . .103
Chapter 7. Continuous nowhere approximately
differentiable functions . . . . . . . . . . . . . . . . . . . . 115
Chapter 8. Blumberg’s theorem and
Sierpin´ski-Zygmund functions . . . . . . . . . . . . . . . . 127
Chapter 9. The cardinality of first Baire class . . . . . . 143
Chapter 10. Lebesgue nonmeasurable functions
and functions without the Baire property . . . . . . . . . 153
Chapter 11. Hamel basis and
Cauchy functional equation . . . . . . . . . . . . . . . . . 175
v
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vi table of contents
Chapter 12. Summation methods and
Lebesgue nonmeasurable functions . . . . . . . . . . . . . 195
Chapter 13. Luzin sets, Sierpin´ski sets,
and their applications . . . . . . . . . . . . . . . . . . . . . 209
Chapter 14. Absolutely nonmeasurable
additive functions . . . . . . . . . . . . . . . . . . . . . . . 229
Chapter 15. Egorov type theorems . . . . . . . . . . . . . 241
Chapter 16. A difference between the Riemann
and Lebesgue iterated integrals . . . . . . . . . . . . . . . 255
Chapter 17. Sierpin´ski’s partition
of the Euclidean plane . . . . . . . . . . . . . . . . . . . . 265
Chapter 18. Bad functions defined on
second category sets . . . . . . . . . . . . . . . . . . . . . 281
Chapter 19. Sup-measurable and weakly
sup-measurable functions . . . . . . . . . . . . . . . . . . 295
Chapter 20. Generalized step-functions and
superposition operators . . . . . . . . . . . . . . . . . . . 313
Chapter 21. Ordinary differential equations
with bad right-hand sides . . . . . . . . . . . . . . . . . . 329
Chapter 22. Nondifferentiable functions from
the point of view of category and measure . . . . . . . . 343
Chapter 23. Absolute null subsets of the plane
with bad orthogonal projections . . . . . . . . . . . . . . 369
Appendix 1. Luzin’s theorem
on the existence of primitives . . . . . . . . . . . . . . . . 383
Appendix 2. Banach limits on the real line . . . . . . . . 391
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table of contents vii
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 401
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
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Preface
At the presenttime, many strange(or singular)objects in various fields
of mathematics are known, and no working mathematician is greatly sur-
prised if he meets some objects of this type during his investigations. In
connection with strange (singular) objects, classical mathematical analysis
must be noticed especially. It is sufficient to recall here the well-known
examples of continuous nowhere differentiable real-valued functions, ex-
amples of Lebesgue measurable real-valued functions nonintegrable on any
nonemptyopensubintervaloftherealline,examplesofLebesgueintegrable
real-valued 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 so-
calledcategorymethodbasedonthecelebratedBairetheoremfromgeneral
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 (also, 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 preciselyof strange(ina certainsense)elements. Many interesting
applicationsofthecategorymethodarepresentedintheexcellenttextbook
byOxtoby[202]inwhichthedeepanalogybetweenmeasureandcategoryis
thoroughlydiscussedaswell. Inthisconnection,themonographbyMorgan
[192] must also be pointed out where an abstract concept generalizing the
notions of measure and category is introduced and investigated in detail.
Unfortunately,thecategorymethoddoesnotalwaysworkandwesome-
times need an essentially different approach to questions concerning the
existence of singular objects.
This book is devoted to some strange functions (and point sets) in real
analysis and their applications. Those functions can be frequently met
duringvariousstudiesinanalysisandplayanessentialrolethere,especially
as counterexamples to numerous statements that at first sight seem to be
very natural but, finally, fail to be true in certain extraordinary situations
(see,e.g.,[78]). Anotherimportantroleofstrangefunctions,withrespectto
ix