Nonmeasurable Sets and Functions

Nonfiction, Science & Nature, Mathematics, Set Theory, Mathematical Analysis
Cover of the book Nonmeasurable Sets and Functions by Alexander Kharazishvili, Elsevier Science
View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart
Author: Alexander Kharazishvili ISBN: 9780080479767
Publisher: Elsevier Science Publication: May 29, 2004
Imprint: Elsevier Science Language: English
Author: Alexander Kharazishvili
ISBN: 9780080479767
Publisher: Elsevier Science
Publication: May 29, 2004
Imprint: Elsevier Science
Language: English

The book is devoted to various constructions of sets which are nonmeasurable with respect to invariant (more generally, quasi-invariant) measures. Our starting point is the classical Vitali theorem stating the existence of subsets of the real line which are not measurable in the Lebesgue sense. This theorem stimulated the development of the following interesting topics in mathematics:

  1. Paradoxical decompositions of sets in finite-dimensional Euclidean spaces;
  2. The theory of non-real-valued-measurable cardinals;
  3. The theory of invariant (quasi-invariant)
    extensions of invariant (quasi-invariant) measures.

These topics are under consideration in the book. The role of nonmeasurable sets (functions) in point set theory and real analysis is underlined and various classes of such sets (functions) are investigated . Among them there are: Vitali sets, Bernstein sets, Sierpinski sets, nontrivial solutions of the Cauchy functional equation, absolutely nonmeasurable sets in uncountable groups, absolutely nonmeasurable additive functions, thick uniform subsets of the plane, small nonmeasurable sets, absolutely negligible sets, etc. The importance of properties of nonmeasurable sets for various aspects of the measure extension problem is shown. It is also demonstrated that there are close relationships between the existence of nonmeasurable sets and some deep questions of axiomatic set theory, infinite combinatorics, set-theoretical topology, general theory of commutative groups. Many open attractive problems are formulated concerning nonmeasurable sets and functions.

· highlights the importance of nonmeasurable sets (functions) for general measure extension problem.
· Deep connections of the topic with set theory, real analysis, infinite combinatorics, group theory and geometry of Euclidean spaces shown and underlined.
· self-contained and accessible for a wide audience of potential readers.
· Each chapter ends with exercises which provide valuable additional information about nonmeasurable sets and functions.
· Numerous open problems and questions.

View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart

The book is devoted to various constructions of sets which are nonmeasurable with respect to invariant (more generally, quasi-invariant) measures. Our starting point is the classical Vitali theorem stating the existence of subsets of the real line which are not measurable in the Lebesgue sense. This theorem stimulated the development of the following interesting topics in mathematics:

  1. Paradoxical decompositions of sets in finite-dimensional Euclidean spaces;
  2. The theory of non-real-valued-measurable cardinals;
  3. The theory of invariant (quasi-invariant)
    extensions of invariant (quasi-invariant) measures.

These topics are under consideration in the book. The role of nonmeasurable sets (functions) in point set theory and real analysis is underlined and various classes of such sets (functions) are investigated . Among them there are: Vitali sets, Bernstein sets, Sierpinski sets, nontrivial solutions of the Cauchy functional equation, absolutely nonmeasurable sets in uncountable groups, absolutely nonmeasurable additive functions, thick uniform subsets of the plane, small nonmeasurable sets, absolutely negligible sets, etc. The importance of properties of nonmeasurable sets for various aspects of the measure extension problem is shown. It is also demonstrated that there are close relationships between the existence of nonmeasurable sets and some deep questions of axiomatic set theory, infinite combinatorics, set-theoretical topology, general theory of commutative groups. Many open attractive problems are formulated concerning nonmeasurable sets and functions.

· highlights the importance of nonmeasurable sets (functions) for general measure extension problem.
· Deep connections of the topic with set theory, real analysis, infinite combinatorics, group theory and geometry of Euclidean spaces shown and underlined.
· self-contained and accessible for a wide audience of potential readers.
· Each chapter ends with exercises which provide valuable additional information about nonmeasurable sets and functions.
· Numerous open problems and questions.

More books from Elsevier Science

Cover of the book Advances in Imaging and Electron Physics by Alexander Kharazishvili
Cover of the book Microwave De-embedding by Alexander Kharazishvili
Cover of the book Negotiating in the Leadership Zone by Alexander Kharazishvili
Cover of the book Vegetable Oil-Based Polymers by Alexander Kharazishvili
Cover of the book Handbook of Pollution Prevention and Cleaner Production Vol. 3: Best Practices in the Agrochemical Industry by Alexander Kharazishvili
Cover of the book Mesoporous Silica-based Nanomaterials and Biomedical Applications - Part A by Alexander Kharazishvili
Cover of the book Studies in Natural Products Chemistry by Alexander Kharazishvili
Cover of the book High-Temperature Solid Oxide Fuel Cells for the 21st Century by Alexander Kharazishvili
Cover of the book Solid State Physics by Alexander Kharazishvili
Cover of the book DW 2.0: The Architecture for the Next Generation of Data Warehousing by Alexander Kharazishvili
Cover of the book Advances in Sheep Welfare by Alexander Kharazishvili
Cover of the book Sustainable Mass Transit by Alexander Kharazishvili
Cover of the book Security Controls Evaluation, Testing, and Assessment Handbook by Alexander Kharazishvili
Cover of the book Contemporary Enzyme Kinetics and Mechanism by Alexander Kharazishvili
Cover of the book Data Mining: Concepts and Techniques by Alexander Kharazishvili
We use our own "cookies" and third party cookies to improve services and to see statistical information. By using this website, you agree to our Privacy Policy