Real analysis

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The book presents a unique approach to measure and integration, beginning with a general abstract framework before focusing on Lebesgue measure and integration on the real line. It comprises nine chapters and an appendix, covering topics from basic set classes and measures to measurable functions and various convergence types. Key concepts include abstract Lebesgue integration, limit theorems, and comparisons between Lebesgue and Riemann integrals. Additionally, it explores Lp spaces, normed vector spaces, and signed measures. The text emphasizes a novel application of Lebesgue theory to enhance understanding of differentiation, broadening the classical total change formula to a wider range of functions. Designed for classroom use, it encourages active student engagement through 125 problems and 358 exercises, which require students to prove statements, fill in proof details, or provide counterexamples. More challenging problems are marked, and many include hints. The book is rich in examples and remarks that clarify definitions and provoke discussion on subtleties and essential conditions. This well-structured text is ideal for a one-semester graduate course in real analysis, appealing to students in mathematics, physics, computer science, and engineering, and preparing them for advanced studies in functional analysis and operator theory.