This book features the interplay of two main branches of mathematics: topology and real analysis. The material of the book is largely contained in the research publications of the authors and their students from the past 50 years. Parts of analysis are touched upon in a unique way, for example, Lebesgue measurability, Baire classes of functions, differentiability, $C^n$ and $C^{infty}$ functions, the Blumberg theorem, bounded variation in the sense of Cesari, and various theorems on Fourier series and generalized bounded variation of a function.

Part 1. The One Dimensional Case 
Subsets of $mathbb{R}$ 
Baire class 1 
Differentiability classes 
The derivative function 
Part 2. Mappings and Measures on ${mathbb R}^n$ 
Bi-Lipschitzian homeomorphisms 
Approximation by homeomorphisms 
Measures on $mathbb{R}^n$ 
Blumberg's theorem 
Part 3. Fourier Series 
Improving the behavior of Fourier series 
Preservation of convergence of Fourier series 
Fourier series of integrable functions 
Supplementary material 
Bibliography 
Index

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