Applications of Point Set Theory in Real Analysis

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Springer Science & Business Media, Mar 31, 1998 - Mathematics - 240 pages
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This book is devoted to some results from the classical Point Set Theory and their applications to certain problems in mathematical analysis of the real line. Notice that various topics from this theory are presented in several books and surveys. From among the most important works devoted to Point Set Theory, let us first of all mention the excellent book by Oxtoby [83] in which a deep analogy between measure and category is discussed in detail. Further, an interesting general approach to problems concerning measure and category is developed in the well-known monograph by Morgan [79] where a fundamental concept of a category base is introduced and investigated. We also wish to mention that the monograph by Cichon, Wę;glorz and the author [19] has recently been published. In that book, certain classes of subsets of the real line are studied and various cardinal valued functions (characteristics) closely connected with those classes are investigated. Obviously, the IT-ideal of all Lebesgue measure zero subsets of the real line and the IT-ideal of all first category subsets of the same line are extensively studied in [19], and several relatively new results concerning this topic are presented. Finally, it is reasonable to notice here that some special sets of points, the so-called singular spaces, are considered in the classi
 

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Contents

1 Setvalued mappings
21
2 Nonmeasurable sets and sets without the Baire property
39
problem
55
4 Some properties of cralgebras and crideals
77
5 Nomneasurable subgroups of the real line
91
the real line
101
7 Translations of sets and functions
111
8 The SteinhaiLS property of invariant measures
123
10 The principle of condensation of singularities
143
measures
161
12 Some subsets of spaces equipped with transformation groups
173
13 Sierpiriskis partition and its applications
185
real line
197
15 Set theory and ordinary differential equations
209
Bibliography
223
Subject Index
233

Luzin
133

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Page viii - Russian version of the book is based on the course of lectures given by the author over a period of 20 years to undergraduate and postgraduate students of the Moscow Power Institute.

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