Measure and Integration
Description
This textbook explores deeply the theory of measure and integration through examples, counterexample, exercises and problem solving. As a first course, the book is designed to provide a self-contained document for graduate students and researchers involved in analysis and probability. The general topics covered by the book are indispensable in understanding most of concepts in functional analysis, harmonic analysis, probability theory, stochastic analysis.
The book does not present the theory of measure and integration only through theorems and proofs. Instead, some important concepts and selected themes are proposed within exercises and problems. This approach is chosen purposely.
A collection of 140 exercises and 20 problems are set out at the end of each chapter with complete solutions. The second part of the textbook offers fully detailed solutions of the proposed problems and exercises.
The challenge was to make the text accessible for any student with a modest background in real analysis without missing the basics and essential results of the theory of measure and integration.
Further to the theory, the textbook offers a large numbers of exercises and problems with detailed solutions. Most solved problems tackle some specific topic not covered by the theoretical part.
The book is practical both as a teaching document for Bachelor, Master students, and first-year PhD students. It is also suitable for advanced undergraduate students and lecturers.
More details
Person
Smail Djebali received a Master degree (1983) and a PhD degree (1987) in Mathematics from Paris XI University (France,) then a Science Doctorate (Habilitation) from Algiers University (2001).
The author is specialized in Mathematical Analysis with a major in Topological methods and Applications to Differential Equations. Yet, the author has skills in Nonlinear Analysis, Applied Analysis (mainly in Integro-Differential Equations and Inclusions), and Algebraic Topology.
The author has supervised 15 PhD theses in this field and more Master projects.
The author has published more than 110 research papers in specialized journals jointly with 45 collaborators, co-authored one Lecture-Notes book (2011) and 3 books (2012, 2023, and 2024).
The author is member of two journal editorial boards and serves as a reviewer for several peer-reviewed journals.
Since 2006, the author acts as a reviewer for Zentrablatt fur Mathematik (European Mathematical Society, www.zentralblatt-math.org/zmath/en/). More than 170 reviews have been performed so far. As of July 2009, the author also acts as a reviewer for Mathematical Reviews (American Mathematical Society, www.ams.org) (more than 80 reviews have been performed).
This textbook evolved from some course lectures that the author has delivered at Master and PhD degree programs for eight academic semesters. This textbook offers a complete and self-contained teaching document for Master and first-year PhD students.