Real Analysis (I)
2021/9/24-2022/1/21 9:10-12:00 on Fridays,
M210, Department of Mathematics, NTNU
Speaker:
Ulrich Menne 孟悟理 (National Taiwan Normal University)
Organizer:
Ulrich Menne 孟悟理 (National Taiwan Normal University)
Background & Purpose
Lebesgue integration theory is one of the pillars of analysis. The present course and its sequel, Real Analysis II, gives a thorough treatment of the underlying general measure theory, Lebesgue integration, the resulting Lebesgue spaces, related linear functionals, and product measures. It treats Borel regular measures, Radon measures, and Riesz’s representation theorem in some depth and includes the theory of Daniell integrals and as well as Riemann-Stieltjes integration. This choice of emphasis facilitates the study of geometric measure theory through planned subsequent courses.
Outline
In part I, after establishing the necessary basics from point-set topology, measures and measurable sets (including numerical summation and measurable hulls), Borel sets (Borel families, the space of sequences of positive integers, images of Borel sets, and Borel functions), Borel regular measures (approximation by closed sets, nonnmeasurable sets, Radon measures, and their images), measurable functions (approximation theorems and spaces of measurable functions), and Lebesgue integration (up to and including limit theorems) shall be treated. In part II in the following term, after establishing the necessary background on tensor products and functional analysis, Lebesgue integration shall be completed (Lebesgue spaces), linear functionals (lattices of functions, Daniell integrals, linear functionals on Lebesgue spaces, Riesz’s representation theorem, curve length, and Riemann-Stieltjes integration), and product measures (Fubini’s theorem and Lebesgue measure) shall be covered. Some of the material on topology and functional analysis will be relegated to the two self-study phases.
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