Analysis and Integration Theory From Riemann to Lebesgue

This book is the culmination of more than two decades of teaching undergraduate and graduate mathematical analysis. It presents a rigorous yet highly accessible treatment of Riemann integration, sequences and series of functions, Fourier series, and measure theory and integration. The exposition emphasizes the development from classical to modern analysis and provides an account of the relationships among different notions of integration and modes of convergence. A central feature of the book is its unified treatment of Riemann and Lebesgue integration within a coherent analytical framework.

The presentation begins with a historical background and proceeds systematically through Riemann integration in one and several variables. This structured progression encourages readers to generalize from Jordan measurable sets to Lebesgue measurable sets, from Jordan measure to Lebesgue measure, and from the Riemann integral to the Lebesgue integral, culminating in three chapters devoted to Lebesgue integration. The treatment of sequences and series of functions stands as a self-contained unit, tracing developments from Euler and Fourier to modern advances.

Key features:

• Rigorous and self-contained development of Riemann integration in R and Rⁿ
  
• Unified treatment of measure theory and Lebesgue integration
• Coverage of multivariable differentiation, including inverse and implicit function theorems
• Detailed treatment of Fourier series, including pointwise, uniform, and L² convergence
• Numerous carefully written examples illustrating key results
• Extensive exercises with hints, supported by a separate solutions manual
• Structured teaching plans together with selected final-year project topics and previous examinations.
• This book is aimed at advanced undergraduate and graduate students in mathematics, as well as PhD students and researchers. Combining mathematical rigor with a clear and systematic presentation, this book serves both as a textbook for courses in analysis and integration theory and as a reference for further study in modern mathematics.

This book is aimed at advanced undergraduate and graduate students in mathematics, as well as PhD students and researchers. Combining mathematical rigor with a clear and systematic presentation, this book serves both as a textbook for courses in analysis and integration theory and as a reference for further study in modern mathematics.