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# Measure and Integration

## Module MA2003

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Module version of SS 2012

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
WS 2020/1SS 2020WS 2012/3SS 2012

### Basic Information

MA2003 is a semester module in German language at Bachelor’s level which is offered in winter semester.

This module description is valid to SS 2021.

150 h 45 h 5 CP

### Content, Learning Outcome and Preconditions

#### Content

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)

#### Learning Outcome

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.

#### Preconditions

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

### Courses, Learning and Teaching Methods and Literature

#### Courses and Schedule

VO 2 Measure and Integration Theory Bornemann, F. Ludwig, C. Mon, 14:00–16:00, Interims I 102
eLearning
documents
UE 1 Measure and Integration Theory (Exercise Session) Bornemann, F. Ludwig, C. dates in groups documents

#### Learning and Teaching Methods

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

#### Literature

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.

### Module Exam

#### Description of exams and course work

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.

#### Exam Repetition

The exam may be repeated at the end of the semester.

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