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: br 1. Paradoxical decompositions of sets in finite-dimensional Euclidean spaces; br 2. The theory of non-real-valued-measurable cardinals; br 3. The theory of invariant (quasi-invariant) br extensions of invariant (quasi-invariant) measures. These topics are under consideration i ..

strong Strange Functions in Real Analysis, Third Edition strong differs from the previous editions in that it includes five new chapters as well as two appendices. More importantly, the entire text has been revised and contains more detailed explanations of the presented material. In doing so, the book explores a number of important examples and constructions of pathological functions. After introducing basic concepts, the author begins with Cantor and Peano-type functions, then moves effortlessly to functions whose constructions require what is essentially non-effective methods. These include functions without the Baire property, functions associated with ..

Weierstrass and Blancmange nowhere differentiable functions, Lebesgue integrable functions with everywhere divergent Fourier series, and various nonintegrable Lebesgue measurable functions. While dubbed strange or "pathological," these functions are ubiquitous throughout mathematics and play an important role in analysis, not only as counterexamples of seemingly true and natural statements, but also to stimulate and inspire the further development of real analysis. Strange Functions in Real Analysis explores a number of important examples and constructions of pathological functions. After introducing the basic concepts, the author begins with Cantor and Peano-t ..

Preface.- 1. Real-Valued Semicontinuous Functions.- 2. The Oscillations of Real-Valued Functions.- 3. Monotone and Continuous Restrictions of Real-Valued Functions.- 4. Bijective Continuous Images of Absolute Null Sets.- 5. Projective Absolutely Nonmeasurable Functions.- 6. Borel Isomorphisms of Analytic Sets.- 7. Iterated Integrals of Real-Valued Functions of Two Real Variables.- 8. The Steinhaus Property, Ergocidity, and Density Points.- 9. Measurability Properties of H-Selectors and Partial H-Selectors.- 10. A Decomposition of an Uncountable Solvable Group into Three Negligible Sets.- 11. Negligible Sets Versus Absolutely Nonmeasurable Sets.- 12. Measurabilit ..