Course: Measure and Integral

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Course title Measure and Integral
Course code KMA/MIM
Organizational form of instruction Lecture + Exercise
Level of course Bachelor
Year of study not specified
Semester Winter
Number of ECTS credits 6
Language of instruction Czech
Status of course Compulsory, Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Vodák Rostislav, RNDr. Ph.D.
  • Pastor Karel, doc. Mgr. Ph.D.
  • Fürst Tomáš, RNDr. Ph.D.
  • Vencálek Ondřej, Mgr. Ph.D.
Course content
1. Definition of measure and sigma-algebras. 2. Basic properties of measures. 3. Outer measure and the Caratheodory extension theorem. 4. The Lebesgue measure. 5. Measurable functions. 6. Sequences of measurable functions and types of convergence. 7. The Lebesgue integral. 8. Properties of the Lebesgue integral. 9. Generalized measures, the Hahn and Jordan decomposition. 10. Radon-Nikodym derivative. 11. The Fubini theorem.

Learning activities and teaching methods
Lecture, Demonstration
  • Attendace - 52 hours per semester
  • Homework for Teaching - 45 hours per semester
  • Preparation for the Exam - 80 hours per semester
Learning outcomes
Understand the abstract construction of an integral based on a measure.
Comprehension Understand the more abstract construction of an integral based on a measure.
Prerequisites
Differential and integral calculus of the functions of several variables.
KMA/MA2M

Assessment methods and criteria
Oral exam

Credit: active participation during seminars. Exam: the student has to understand the subject and be able to prove all theorems.
Recommended literature
  • J. Lukeš, J. Malý. (1995). Measure and Intergral. Matfyzpress, Praha.
  • P. R. Halmos. (1950). Measure theory. New York, D. Van Nostrand Company.
  • V. Jarník. (1984). Integrální počet (I), (II).. Academia, Praha.


Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Science Applications of Mathematics in Economy (2015) Mathematics courses 1 Winter
Faculty of Science Mathematics and Applications (1) Mathematics courses 2 Winter
Faculty of Science General Physics and Mathematical Physics (1) Physics courses 2 Winter