Ebook: Applications of Point Set Theory in Real Analysis

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, W«;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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The main goal of this book is to demonstrate the usefulness of set-theoretical methods in various questions of real analysis and classical measure theory. In this context, many statements and facts from analysis are treated as consequences of purely set-theoretical assertions which can successfully be applied to measures and Baire category. Topics covered include similarities and differences between measure and category; constructions of nonmeasurable sets and of sets without the Baire property; three aspects of the measure extension problem; the principle of condensation of singularities from the point of view of the Kuratowski-Ulam theorem; transformation groups and invariant (quasi-invariant) measures; the uniqueness property of an invariant measure; and ordinary differential equations with nonmeasurable right-hand sides.
Audience: The material presented in the book is essentially self-contained and is accessible to a wide audience of mathematicians. It will appeal to specialists in set theory, mathematical analysis, measure theory and general topology. It can also be recommended as a textbook for postgraduate students who are interested in the applications of set-theoretical methods to the above-mentioned domains of mathematics.

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The main goal of this book is to demonstrate the usefulness of set-theoretical methods in various questions of real analysis and classical measure theory. In this context, many statements and facts from analysis are treated as consequences of purely set-theoretical assertions which can successfully be applied to measures and Baire category. Topics covered include similarities and differences between measure and category; constructions of nonmeasurable sets and of sets without the Baire property; three aspects of the measure extension problem; the principle of condensation of singularities from the point of view of the Kuratowski-Ulam theorem; transformation groups and invariant (quasi-invariant) measures; the uniqueness property of an invariant measure; and ordinary differential equations with nonmeasurable right-hand sides.
Audience: The material presented in the book is essentially self-contained and is accessible to a wide audience of mathematicians. It will appeal to specialists in set theory, mathematical analysis, measure theory and general topology. It can also be recommended as a textbook for postgraduate students who are interested in the applications of set-theoretical methods to the above-mentioned domains of mathematics.
Content:
Front Matter....Pages i-viii
Introduction: preliminary facts....Pages 1-20
Set-valued mappings....Pages 21-38
Nonmeasurable sets and sets without the Baire property....Pages 39-54
Three aspects of the measure extension problem....Pages 55-76
Some properties of ?-algebras and ?-ideals....Pages 77-90
Nonmeasurable subgroups of the real line....Pages 91-100
Additive properties of invariant ?-ideals on the real line....Pages 101-110
Translations of sets and functions....Pages 111-122
The Steinhaus property of invariant measures....Pages 123-132
Some applications of the property (N) of Luzin....Pages 133-142
The principle of condensation of singularities....Pages 143-160
The uniqueness of Lebesgue and Borel measures....Pages 161-172
Some subsets of spaces equipped with transformation groups....Pages 173-184
Sierpi?ski’s partition and its applications....Pages 185-196
Selectors associated with subgroups of the real line....Pages 197-208
Set theory and ordinary differential equations....Pages 209-222
Back Matter....Pages 223-240

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The main goal of this book is to demonstrate the usefulness of set-theoretical methods in various questions of real analysis and classical measure theory. In this context, many statements and facts from analysis are treated as consequences of purely set-theoretical assertions which can successfully be applied to measures and Baire category. Topics covered include similarities and differences between measure and category; constructions of nonmeasurable sets and of sets without the Baire property; three aspects of the measure extension problem; the principle of condensation of singularities from the point of view of the Kuratowski-Ulam theorem; transformation groups and invariant (quasi-invariant) measures; the uniqueness property of an invariant measure; and ordinary differential equations with nonmeasurable right-hand sides.
Audience: The material presented in the book is essentially self-contained and is accessible to a wide audience of mathematicians. It will appeal to specialists in set theory, mathematical analysis, measure theory and general topology. It can also be recommended as a textbook for postgraduate students who are interested in the applications of set-theoretical methods to the above-mentioned domains of mathematics.
Content:
Front Matter....Pages i-viii
Introduction: preliminary facts....Pages 1-20
Set-valued mappings....Pages 21-38
Nonmeasurable sets and sets without the Baire property....Pages 39-54
Three aspects of the measure extension problem....Pages 55-76
Some properties of ?-algebras and ?-ideals....Pages 77-90
Nonmeasurable subgroups of the real line....Pages 91-100
Additive properties of invariant ?-ideals on the real line....Pages 101-110
Translations of sets and functions....Pages 111-122
The Steinhaus property of invariant measures....Pages 123-132
Some applications of the property (N) of Luzin....Pages 133-142
The principle of condensation of singularities....Pages 143-160
The uniqueness of Lebesgue and Borel measures....Pages 161-172
Some subsets of spaces equipped with transformation groups....Pages 173-184
Sierpi?ski’s partition and its applications....Pages 185-196
Selectors associated with subgroups of the real line....Pages 197-208
Set theory and ordinary differential equations....Pages 209-222
Back Matter....Pages 223-240
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