A uniquely accessible book for general measure and integration, emphasizing the real line, Euclidean space, and the underlying role of translation in real analysis

"Measure and Integration: A Concise Introduction to Real Analysis" presents the basic concepts and methods that are important for successfully reading and understanding proofs. Blending coverage of both fundamental and specialized topics, this book serves as a practical and thorough introduction to measure and integration, while also facilitating a basic understanding of real analysis.

The author develops the theory of measure and integration on abstract measure spaces with an emphasis of the real line and Euclidean space. Additional topical coverage includes:

Measure spaces, outer measures, and extension theorems

Lebesgue measure on the line and in Euclidean space

Measurable functions, Egoroff's theorem, and Lusin's theorem

Convergence theorems for integrals

Product measures and Fubini's theorem

Differentiation theorems for functions of real variables

Decomposition theorems for signed measures

Absolute continuity and the Radon-Nikodym theorem

Lp spaces, continuous-function spaces, and duality theorems

Translation-invariant subspaces of L2 and applications

The book's presentation lays the foundation for further study of functional analysis, harmonic analysis, and probability, and its treatment of real analysis highlights the fundamental role of translations. Each theorem is accompanied by opportunities to employ the concept, as numerous exercises explore applications including convolutions, Fourier transforms, and differentiation across the integral sign.

Providing an efficient and readable treatment of this classical subject, "Measure and Integration: A Concise Introduction to Real Analysis" is a useful book for courses in real analysis at the graduate level. It is also a valuable reference for practitioners in the mathematical sciences.

Features an introduction to advanced calculus and highlights its inherent concepts from linear algebra

"Advanced Calculus" reflects the unifying role of linear algebra in an effort to smooth readers' transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting a sound approach to the study. The traditional theorems of elementary differential and integral calculus are rigorously established, presenting the foundations of calculus in a way that reorients thinking toward modern analysis.

Following an introduction dedicated to writing proofs, the book is divided into three parts:

Part One explores foundational one-variable calculus topics from the viewpoint of linear spaces, norms, completeness, and linear functionals.

Part Two covers Fourier series and Stieltjes integration, which are advanced one-variable topics.

Part Three is dedicated to multivariable advanced calculus, including inverse and implicit function theorems and Jacobian theorems for multiple integrals.

Numerous exercises guide readers through the creation of their own proofs, and they also put newly learned methods into practice. In addition, a "Test Yourself" section at the end of each chapter consists of short questions that reinforce the understanding of basic concepts and theorems. The answers to these questions and other selected exercises can be found at the end of the book along with an appendix that outlines key terms and symbols from set theory.

Guiding readers from the study of the topology of the real line to the beginning theorems and concepts of graduate analysis, "Advanced Calculus" is an ideal text for courses in advanced calculus and introductory analysis at the upper-undergraduate and beginning-graduate levels. It also serves as a valuable reference for engineers, scientists, and mathematicians.