Measure and Integration

sigma algebra, measure, Borel sigma algebra, Lebesgue measure,
integration theory in Rn based on the Lebesgue integral (including Fubini's theorem and transformation formula)
extension and uniqueness for sigma additive set functions (without proof)
measurable mappings, image measure, absolute continuity, density, general Lebesgue integral;
monotone and dominated convergence, Fatou's lemma, Lp spaces (including the Hölder and Minkowski inequalities, completeness)

Having completed the course with success, the student is able to handle with confidence the Lebesgue integral in a general measure theoretic context and with respect to its fundamental convergence properties.

MA1001 Analysis 1, MA1002 Analysis 2, MA1101 Linear Algebra and Discrete Structures 1, MA1102 Linear Algebra and Discrete Structures 2

The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should animate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress. At the beginning of the module, the practice sessions will be offered under guidance, but during the term the sessions will become more independent, and intensify learning individually as well as in small groups.

blackboard

D. Werner, Kapitel IV aus: Einführung in die höhere Analysis. Springer, 2006.
E.H. Lieb, M. Loss, Chapter 1 and 2 from: Analysis. American Mathematical Society, 2nd edition, 2001.
M. Brokate, G. Kersting: Maß und Integral. Birkhäuser, 2010. English translation: Measure and integral. Birkhäuser, 2015.

The module examination is based on a written exam (60 minutes). Students have to deal with Lebesgue integrals in a general measure theoretical context adequately and determine the basic convergence properties.