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Lecturer(s)
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Soudský Filip, RNDr. Ph.D.
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Course content
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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.
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Learning activities and teaching methods
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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
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Learning outcomes
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The subject is an introduction to measure and integral theory. The student will become familiar with abstract spaces with measure and subsequently with Lebesgue outer measure and Lebesgue measure. Subsequently, the Lebesgue integral over an abstract space with measure is defined and its basic properties are derived (monotonicity, linearity, limit theorems (Fatou's, Levi's and Lebesgue's), Fubini's theorem). The student will also become familiar with Lebesgue function spaces and their basic properties (completeness, reflexivity, etc...). Next, basic relations between L^p norms, H?older's inequality and dual spaces to L^p (Riesz's theorem) are discussed.
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.
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Prerequisites
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Calculus 1, Calculus 2, Analysis of Functions of Several Variabl
KMA/PAN1M and KMA/PAN2M
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Assessment methods and criteria
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Combined examination
Credit: Active participation on seminars + tests. Exam: writtten and oral.
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Recommended literature
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Jirásek, F. - Čipera, S. - Vacek, M.:. Sbírka řešených příkladů z matematiky II. SNTL, Praha, 1989.
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Brabec, J. - Hrůza, B.:. Matematická analýza II. Praha, 1986.
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Jarník, V.:. Integrální počet II. Praha, ČSAV 1955.. ČSAV, Praha, 1955.
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Lang, S,:. Real and Functional Analysis. Springer Verlag, New York, 1993.
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Lukeš, J.:. Příklady z matematické analýzy I. Příklady k teorii Lebesgueova integrálu. MFF UK Praha, 1968.
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Lukeš, L. - Malý, J.:. Míra a integrál. [skripta MFF UK], Praha, UK 1993.. skripta MFF UK, Praha, 1993.
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Netuka, I. - Veselý, J.:. Příklady z matematické analýzy. Míra a integrál. skripta MFF UK Praha, 1982.
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Royden, H. L.:. Real analysis. New York, The Macmillan Company 1963.. The Macmillan Company, New York, 1963.
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Rudin, W.:. Analýza v reálném a komplexním oboru. Praha, Academia 1977.. Academia, Praha, 1977.
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Sikorski, R.:. Diferenciální a integrální počet. Praha, Academia 1973.. Academia, Praha, 1973.
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