Young measures and compactness in measure spaces

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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.