This book offers several topics of mathematical analysis which are closely connected with significant properties of real-valued functions of various types (such as semi-continuous functions, monotone functions, convex functions, measurable functions, additive and linear functionals, etc.). Alongside with fairly traditional themes of real analysis and classical measure theory, more profound questions are thoroughly discussed in the book – appropriate extensions and restrictions of functions, oscillation functions and their characterization, discontinuous functions on resolvable topological spaces, pointwise limits of finite sums of periodic functions, some general results on invariant and quasi-invariant measures, the structure of non-measurable sets and functions, the Baire property of functions on topological spaces and its connections with measurability properties of functions, logical and set-theoretical aspects of the behavior of real-valued functions.
Chapter
1. Unary and Binary Relations.
Chapter
2. Partial Functions and
Functions.
Chapter
3. Elementary Facts on Cardinal Numbers.
Chapter
4. Some
Properties of the Continuum.
Chapter
5. The Oscillation of a Real-valued
Function at a Point.
Chapter
6. Points of Continuity and Discontinuity of
Real-valued Functions.
Chapter
7. Real-valued Monotone Functions.
Chapter
8. Real-valued Convex Functions.
Chapter
9. Semicontinuity of a Real-valued
Function at a Point.
Chapter
10. Semicontinuous Real-valued Functions on
Quasi-compact Spaces.
Chapter
11. The BanachSteinhaus Theorem.
Chapter
12.
A Characterization of Oscillation Functions.
Chapter
13. Semicontinuity
versus Continuity.
Chapter
14. The Outer Measures.
Chapter
15. Finitely
Additive and Countably Additive Measures.
Chapter
16. Extensions of
Measures.
Chapter
17. Caratheodorys and Marczewskis Extension Theorems.-
Chapter
18. Positive Linear Functionals.
Chapter
19. The Nonexistence of
Universal Countably Additive Measures.
Chapter
20. Radon Measures.
Chapter
21. Invariant and Quasi-invariant Measures.
Chapter
22. Pointwise Limits of
Finite Sums of Periodic Functions.
Chapter
23. Absolutely Nonmeasurable
Setsin Commutative Groups.
Chapter
24. Radon Spaces.
Chapter
25.
Nonmeasurable Sets with respect to Radon Measures.
Chapter
26. The
RadonNikodym Theorem.
Chapter
27. Decompositions of Linear Functionals.-
Chapter
28. Linear Continuous Functionals and Radon Measures.
Chapter 29.
Linear Continuous Functionalson a Real Hilbert Space.
Chapter 30. Baire
Property in Topological Spaces.
Chapter 31. The StoneWeierstrass Theorem.-
Chapter 32. More on the Function Space C(X).
Chapter 33. Uniformization of
Plane Sets by Relatively Measurable Functions.
Alexander Kharazishvili is a chief researcher at the A. Razmadze Mathematical Institute of Tbilisi State University and a member of the Georgian National Academy of Sciences. His research interests mainly concern real analysis and measure theory, mostly with various properties of real-valued functions such as topological, algebraic, measure-theoretical, etc. He has more than 300 scientific publications and is the author of the book "Strange Functions in Real Analysis", published by CRC Press. The third edition of this book was published in 2018.
