Essentials of Measure Theory

This textbook serves as a foundational resource for measure theory, suitable for both graduate students and advanced undergraduates, covering essential concepts and applications.

This textbook offers a classical approach to measure theory, divided into two comprehensive parts. The first part is tailored for a one-semester introductory graduate course, focusing on an abstract perspective of measure and integration. Here, the foundational concepts of Lebesgue measure and Lebesgue integral are explored as significant instances of the broader theoretical framework. This structured approach not only lays the groundwork for understanding measure theory but also encourages students to grasp its applications in various contexts.

Part II of the textbook caters to a more advanced audience, presenting material suitable for a subsequent one-semester graduate course. This section delves into measure and integration within topological spaces, expanding on the concepts introduced in the initial part. Each chapter is meticulously designed, culminating in sections that challenge students with problems integral to the learning process. These problems often include theoretical components, accompanied by helpful hints to guide students through complex ideas.

Essentials of Measure Theory is crafted with modest prerequisites, making it accessible to mathematics students while also appealing to those in fields such as statistics, economics, engineering, and physics. The first part may even be suitable for advanced undergraduates who have completed an introductory course in analysis, linear algebra, and elementary set theory. Additionally, the book features suggested readings at the end of each chapter, promoting a deeper understanding of alternative approaches and results.