Measures, Integrals and Martingales

Measures, Integrals and Martingales offers a clear and accessible introduction to measure and integration theory, a cornerstone in areas such as analysis, probability, mathematical physics, and finance. In this second edition, the essential concepts of measure and integration are robustly developed, with martingales providing a natural progression into the deeper aspects of the theory. The text covers fundamental topics including Jacobi's transformation theorem, the Radon–Nikodym theorem, differentiation of measures, and the Hardy–Littlewood maximal functions. This updated edition introduces new chapters on Hausdorff measures, Fourier analysis, vague convergence, as well as classical proofs of both the Radon–Nikodym and Riesz representation theorems. Every proof is presented in detail to ensure a thorough understanding of the material. Designed with few prerequisites in mind, the book is well-suited for undergraduate courses or self-study, and is enriched with numerous illustrations and over 400 exercises. Full solutions for all exercises are available on the author's webpage at www.motapa.de.