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Course: Measure and Integral

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Course title Measure and Integral
Course code KMA/TEM
Organizational form of instruction Lecture
Level of course Bachelor
Year of study not specified
Semester Winter
Number of ECTS credits 3
Language of instruction Czech
Status of course Compulsory
Form of instruction Face-to-face
Work placements Course does not contain work placement
Recommended optional programme components None
Course availability The course is available to visiting students
Lecturer(s)
  • Brzezina Miroslav, doc. RNDr. CSc., dr. h. c.
  • Šimůnková Martina, RNDr. Ph.D.
Course content
Lectures: 1. Basic notions of measure theory: sigma algebra, measure space, measurable functions, steps functions. 2. Integral of step functions, the L1 -completion. 3. Properties of integral. 4. Exchange of the limit and the integral: Fatou's lemma, Levi's and Lebesg's theorems (monotone convergence, dominated convergence). 5. Extension of measures from algebras to sigma algebras. 6. Product measures and Fubini's theorem. 7. Integral and measures in R, relations of the Lebesgue, Newton and Riemann integrals. 8. Distribution functions, the Lebesgue-Stieltjes measure. 9. Lebesgue's measure and integral in Rn. 10. Change of variable formula. 11. Curves, orientation. 12. Curve integrals of the first and second kinds. 13. Green's theorem. Independence of the curve integral on the path. 14. Reserve. Exercises are devoted to practise the subject introduced at the last week lecture.

Learning activities and teaching methods
Monological explanation (lecture, presentation,briefing)
  • Class attendance - 56 hours per semester
  • Preparation for credit - 28 hours per semester
  • Preparation for exam - 28 hours per semester
  • Home preparation for classes - 68 hours per semester
Learning outcomes
Basic properties of measure theory and abstract integration on measure spaces. These are convenient tools for a futher study of mathematical analysis, probability theory and applications.
Basic properties of measure theory and abstract integration on measure spaces. These are convenient tools for a futher study of mathematical analysis, probability theory and applications.
Prerequisites
Calculus 1, Calculus 2, Analysis of Functions of Several Variabl

Assessment methods and criteria
Combined examination

Credit: Active participation on seminars + tests. Exam: writtten and oral.
Recommended literature
  • Jirásek, F. - Čipera, S. - Vacek, M.:. Sbírka řešených příkladů z matematiky II. SNTL, Praha, 1989.
  • Brabec, J. - Hrůza, B.:. Matematická analýza II. Praha, 1986.
  • Jarník, V.:. Integrální počet II. Praha, ČSAV 1955.. ČSAV, Praha, 1955.
  • Lang, S,:. Real and Functional Analysis. Springer Verlag, New York, 1993.
  • Lukeš, J.:. Příklady z matematické analýzy I. Příklady k teorii Lebesgueova integrálu. MFF UK Praha, 1968.
  • Lukeš, L. - Malý, J.:. Míra a integrál. [skripta MFF UK], Praha, UK 1993.. skripta MFF UK, Praha, 1993.
  • Netuka, I. - Veselý, J.:. Příklady z matematické analýzy. Míra a integrál. skripta MFF UK Praha, 1982.
  • Royden, H. L.:. Real analysis. New York, The Macmillan Company 1963.. The Macmillan Company, New York, 1963.
  • Rudin, W.:. Analýza v reálném a komplexním oboru. Praha, Academia 1977.. Academia, Praha, 1977.
  • Sikorski, R.:. Diferenciální a integrální počet. Praha, Academia 1973.. Academia, Praha, 1973.


Study plans that include the course
Faculty Study plan (Version) Category of Branch/Specialization Recommended year of study Recommended semester
Technical University of Liberec, date of update: 02.05.2024 23:50.