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MU:MU17001 Measure Theory and Integration - Course Information

## MU17001 Measure Theory and Integration

**Mathematical Institute in Opava**

Winter 2020

**Extent and Intensity**- 2/2/0. 6 credit(s). Type of Completion: zk (examination).
**Teacher(s)**- doc. RNDr. Michaela Mlíchová, Ph.D. (lecturer)

doc. RNDr. Zdeněk Kočan, Ph.D. (seminar tutor) **Guaranteed by**- doc. RNDr. Michaela Mlíchová, Ph.D.

Mathematical Institute in Opava **Timetable**- Wed 9:45–11:20 118
- Timetable of Seminar Groups:

*Z. Kočan* **Prerequisites**(in Czech)- TYP_STUDIA ( B )
**Course Enrolment Limitations**- The course is also offered to the students of the fields other than those the course is directly associated with.
**fields of study / plans the course is directly associated with**- Mathematical Methods and Modelling (programme MU, Bc-M)
- General Mathematics (programme MU, Bc-M)

**Course objectives**- The main aim of this course is to provide an introduction to the theory of measures and integration with emphasis on Lebesgue measure and Lebesgue integral.
**Syllabus**- 1. Abstract measure theory

- Sigma-albegra and related structures

- measure, complete measure

- outer measure, Carathéodory theorem

- Hopf's extension theorem of measure

2. Lebesgue measure on R^n

- construction of the Lebesgue measure

- metric outer measure

- measurable and non-measurable sets

3. Measurable function

- simple measurable functions

- approximation of measurable function by simple function

- sequences of measurable functions, Jegorov theorem

4. Abstract integration theory

- integrals of a simple measurable function

- integrals of a measurable functions

- basic properties

- Relationships to Riemann and Lebesgue integrals

- 1. Abstract measure theory
**Literature**- A. M. Bruckner, J. B. Bruckner, B. S. Thomson.
*Real Analysis*. Upper Saddle River, New Jersey, 1997. ISBN 0-13-458886-X. info - M. Švec, T. Šalát, T. Neubrunn.
*Matematická analýza funkcií reálnej premennej*. Bratislava, 1987. info

*required literature*- Gail S. Nelson.
*A user-friendly introduction to Lebesgue measure and integration*. American Mathematical Society, 2015. ISBN 978-1-4704-2199-1. info

*recommended literature*- A. M. Bruckner, J. B. Bruckner, B. S. Thomson.
**Language of instruction**- Czech
**Further comments (probably available only in Czech)**- Study Materials

The course can also be completed outside the examination period. **Teacher's information**- Requirements for pre-exam credits are active participation in tutorials and individual presentation of at least two written homework assignments.

The examination is oral and verifies professional knowledge and skills resulting from the course.

- Enrolment Statistics (recent)

- Permalink: https://is.slu.cz/course/sumu/winter2020/MU17001