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Measure and Integration Theory I (O) - NMAA169

Title:	Teorie míry a integrálu I (O)
Guaranteed by:	Department of Mathematical Analysis (32-KMA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2022
Semester:	winter
E-Credits:	3
Hours per week, examination:	winter s.:2/0, Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	cancelled
Language:	Czech
Teaching methods:	full-time
Teaching methods:	full-time
Is provided by:	NMAA069

Classification:	Mathematics > Real and Complex Analysis
Incompatibility :	NMAA069

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Annotation -

Last update: G_M (27.04.2012)

Introductory course on measure theory and integration. Relations between various definitions of the integral; techniques of integral calculus.

Aim of the course -

Last update: G_M (27.04.2012)

Abstract integration and measure theory as a basis for the study of modern mathematical analysis and probability theory.

Literature - Czech

Last update: G_M (27.04.2012)

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

Teaching methods -

Last update: G_M (27.04.2012)

lecture and exercises

Syllabus -

Last update: G_M (27.04.2012)

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.