Real Analysis (Ⅱ)
2022/2/18-2022/6/17 9:10-12:00 on Fridays,
M210, Department of Mathematics, NTNU
Speaker:
Ulrich Menne 孟悟理 (National Taiwan Normal University)
Organizer:
Nan-Kuo Ho 何南國 (National Tsing Hua University)
Background & Purpose
Lebesgue integration theory is one of the pillars of analysis. The present course and its prequel, Real Analysis I, give a thorough treatment of the underlying general measure theory, Lebesgue integration, the resulting Lebesgue spaces, related linear functionals, and product measures. They treat 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, locally compact Hausdorff spaces, 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), and measurable functions (approximation theorems and spaces of measurable functions) were treated. In the present part II, after establishing the necessary background on tensor products and functional analysis, Lebesgue integration (limit theorems and 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 functional analysis will be relegated to the self-study phase.
Details of the course
The main reference text will be the instructor’s weekly updated lecture notes written in LaTeX. They are based on and expand the relevant parts of Federer’s treatise [Fed69] and, concerning topology, Kelley’s book [Kel75]. Grading is solely determined by individual oral examinations conducted in English; to be admitted to these examinations, at least 50% of the possible credits need to be obtained in weekly exercises. The course will be accompanied by a tutorial conducted in Mandarin to review these exercises and to provide assistance with the study of the course and repetition of relevant prerequisites (see below). The course is conducted in the format 16 weeks of lectures plus 2 weeks of self-study.
Prerequisites
Part II assumes basic knowledge of locally compact Hausdorff spaces and a good knowledge of general measure theory; the lecture notes of Part I (see [Men22]) cover this material.