## Applications of Point Set Theory in Real AnalysisThis 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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### 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 |

223 | |

233 | |

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algebra applying arbitrary assertion Axiom of Choice Baire property Banach basic set Borel measure Borel subset card(X category subset closed subset commutative group Continuum Hypothesis countable chain condition countable family element equality equivalent Euclidean example F-selector fact family of sets finite formulate G-invariant G-quasiinvariant graph group G Haar measure Hamel basis i-measurable implies inequality invariant measures ISBN Kuratowski Lebesgue measure Lebesgue nonmeasurable Lebesgue sense Lemma let f let G Let us put Let us recall Luzin mapping acting mapping F Martin's Axiom measurable cardinal measure defined measure space metric space natural number nonempty nonmeasurable with respect nonzero r-finite Obviously ordinal number partial mapping partition Polish topological space product space Proposition proved r-algebra of subsets r-finite measure r-ideal of subsets rational numbers real line real-valued result satisfying the following selector set-valued mapping Sierpiriski strictly positive Suppose theory ZF topological group topological space transfinite recursion uncountable vector space zero

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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.