Measures, Integrals and Martingales
Cambridge University Press, 03.04.2017 - 490 Seiten
A concise yet elementary introduction to measure and integration theory, which are vital in many areas of mathematics, including analysis, probability, mathematical physics and finance. In this highly successful textbook, core ideas of measure and integration are explored, and martingales are used to develop the theory further. Other topics are also covered such as Jacobi's transformation theorem, the Radon-Nikodym theorem, differentiation of measures and Hardy-Littlewood maximal functions. In this second edition, readers will find newly added chapters on Hausdorff measures, Fourier analysis, vague convergence and classical proofs of Radon-Nikodym and Riesz representation theorems. All proofs are carefully worked out to ensure full understanding of the material and its background. Requiring few prerequisites, this book is suitable for undergraduate lecture courses or self-study. Numerous illustrations and over 400 exercises help to consolidate and broaden knowledge. Full solutions to all exercises are available on the author's webpage at www.motapa.de. This book forms a sister volume to René Schilling's other book Counterexamples in Measure and Integration (www.cambridge.org/9781009001625).
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Assume Borel sets compact sets continuous functions convex Corollary countable CP(pl defined Definition denote disjoint dominated convergence theorem Dynkin system e.gy Example exists finite measure space Fourier transform Hausdorff Hint Hölder's inequality increasing sequence inequality interval Lebesgue integral Lebesgue measure Lebesgue's Lemma Let u e Levi's theorem lim Sup limit limn martingale measurable functions measure pu measure space metric space monotone neN neN null sets numbers o-algebra o-finite open sets orthogonal pl(A polynomials probability space Problem proof of Theorem properties proves Radon–Nikodym rectangles resp Riemann integrable satisfies Show simple functions ſº Step sub-o-algebra submartingale ſudu ſun Theorem 6.1 uldu un(x un)nen uniformly integrable unique