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Introductory course on measure theory and integration. Relations between various definitions of the integral; techniques of
integral calculus.
Last update: T_MUUK (21.04.2008)
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Abstract integration and measure theory as a basis for the study of modern mathematical analysis and probability theory. Last update: T_MUUK (21.04.2008)
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W. Rudin: Analýza v reálném a komplexním oboru, Academia, Praha, 2003
J. Lukeš, J. Malý: Míra a integrál (Measure and integral), skripta MFF
J. Kopáček: Matematická analýza pro fyziky III, skripta MFF
J. Lukeš: Příklady z matematické analýzy I. Příklady k teorii Lebesgueova integrálu, skripta MFF
I. Netuka, J. Veselý: Příklady z matematické analýzy. Míra a integrál, skripta MFF
Last update: T_MUUK (24.04.2008)
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lecture and exercises Last update: T_MUUK (28.04.2008)
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1. Basic notions of measure theory.
Sigma - algebra, Borel sets, measure, complete measure, measurable functions, simple functions.
2. Lebesgue measure in R^n.
Existence and uniqueness of Lebesgue measure and its properties.
3. Abstract integral.
Construction of integral on a measure space. Fatou's lemma, Levi's and Lebesgue's theorems (monotone convergence, dominated convergence). Chebyshev's inequality.
4. Integral and measure in R.
Relation of the Lebesgue, Newton and Riemann integrals. Distribution functions and Lebesgue-Stieltjes measure.
5. L^p spaces and convergence of sequences of functions.
Almost everywhere convergence, Jegorov's theorem.
6. Measure theory.
Image of a measure. Radon - Nikodym theorem and Lebesgue's decomposition. Signed measures. Hahn and Jordan decomposition. Last update: T_MUUK (28.04.2008)
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