Liviu C. Florescu Libri

Focusing on accessibility, this textbook simplifies the complex theory of measure and integration, particularly the Lebesgue integral. It caters to less experienced readers by minimizing prerequisites and challenges typically found in other texts. The approach aims to provide a clear understanding of the fundamental concepts and properties, making it an essential resource for those new to the subject. Its compact format ensures that learners can grasp the material without feeling overwhelmed.

The book explores four unique topics relevant to undergraduate courses that are often overlooked in standard lectures. It aims to enrich students' understanding and provide additional insights into these areas, enhancing their overall educational experience.

Recent technological advancements have increased the demand for complex mathematical models, particularly in optimization theory, where optimal solutions are often elusive. Non-convex optimization problems, common in variational calculus and optimal control, may lack classical minimizers due to rapid oscillations in minimizing sequences. This necessitates a relaxation of the solution concept, often achieved through Young measures. This monograph serves as a comprehensive resource, detailing the theoretical aspects of Young measures, including measurability, disintegration, stable convergence, and compactness. It also acts as a valuable reference for those exploring the foundations of measure theory. The text is organized into three chapters: the first provides foundational material on measure theory within an abstract framework; the second focuses on measure theory in topological spaces; and the third applies the compactness results from the first two chapters to the study of Young measures. Each result is thoroughly demonstrated, with multiple proofs for many statements, ensuring that all claims are fully justified and validated.