• Authors: S. Kumaresan & Dharmatti Sheetal
• ISBN: 9789389314878
• Pages: 250
• Language: English
• Publisher: Techno World

About the Book

A Course in Lebesgue Measure & Integration is a comprehensive textbook that introduces the fundamental concepts of Measure Theory and Lebesgue Integration in a clear, intuitive, and mathematically rigorous manner. Designed primarily for postgraduate students of Mathematics, the book follows the standard curriculum prescribed by universities while making the subject accessible through geometric intuition, detailed explanations, and carefully structured proofs.

Measure Theory forms the foundation of advanced mathematical analysis and plays a vital role in areas such as Functional Analysis, Probability Theory, Real Analysis, Harmonic Analysis, and Differential Equations. This book bridges the gap between abstract theory and conceptual understanding by reviving the geometric ideas that originally inspired Henri Lebesgue's development of measure theory.

Beginning with the extension of the notion of length to arbitrary subsets of the real line, the text gradually develops concepts such as outer measure, measurable sets, measurable functions, Lebesgue measure, and Lebesgue integration. Special emphasis is placed on various modes of convergence of functions, supported by numerous examples, practical insights, and guidance on selecting appropriate convergence theorems in different situations.

The book also presents a detailed comparison between Riemann and Lebesgue integration, helping readers understand the advantages, limitations, and applications of each approach. Technical proofs are explained with precision and supported by illustrations wherever appropriate, making even complex concepts easier to visualize and comprehend.

Rich with in-text exercises, end-of-chapter problems, hints, and optional advanced sections titled "For a Good Measure", this book is an excellent resource for students preparing for university examinations, higher studies, research, and competitive examinations requiring a strong background in Real Analysis and Measure Theory.

Key Features

• Comprehensive introduction to Measure Theory and Lebesgue Integration
• Based on intuitive geometric foundations for better conceptual understanding
• Clear explanation of outer measure, measurable sets, and measurable functions
• Detailed treatment of Lebesgue Measure and Lebesgue Integration
• Extensive discussion on different modes of convergence of functions
• Comparative study of Riemann and Lebesgue Integration
• Fully explained proofs with mathematical rigor
• Illustrations and diagrams supporting theoretical concepts
• Numerous solved examples and explanatory notes
• In-text exercises and end-of-chapter practice problems with hints
• Special advanced sections titled "For a Good Measure"
• Suitable for postgraduate studies, research scholars, and advanced mathematics learners

Ideal For

• M.Sc. Mathematics Students
• Postgraduate Mathematics Programmes
• University Students Studying Real Analysis
• Research Scholars in Mathematics
• Teachers of Mathematical Analysis
• Students Preparing for NET, SET, GATE, and Other Higher Studies
• Learners Interested in Measure Theory and Functional Analysis