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Measure and Integration Theory - NMMA203

Title:	Teorie míry a integrálu
Guaranteed by:	Department of Mathematical Analysis (32-KMA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2016 to 2016
Semester:	winter
E-Credits:	8
Hours per week, examination:	winter s.:4/2, C+Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech
Teaching methods:	full-time
Teaching methods:	full-time

Guarantor:	prof. RNDr. Jan Malý, DrSc.
Class:	M Bc. MMIB M Bc. MMIB > Povinné M Bc. MMIB > 2. ročník M Bc. OM M Bc. OM > Povinné M Bc. OM > 2. ročník
Classification:	Mathematics > Real and Complex Analysis
Incompatibility :	{Old courses on Measure Theory I and II}
Pre-requisite :	{One 1st year Analysis course}
Interchangeability :	{Old courses on Measure Theory I and II}
Is co-requisite for:	NMSA202
Is incompatible with:	NMMA903
Is pre-requisite for:	NMMA331, NMMA301
Is interchangeable with:	NMAA070, NMMA903, NMAA069

Opinion survey results Examination dates Schedule Noticeboard

Annotation -

Last update: G_M (16.05.2012)

Introductory course on measure theory and integration. Required course for bachelor's programs General Mathematics and Information Security.

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: prof. RNDr. Jan Malý, DrSc. (07.11.2018)

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: prof. RNDr. Jan Malý, DrSc. (05.11.2013)

1. Basic notions of measure theory.

a) Sigma-albegra and related structures, measure

b) Measurable functions

2. Construction of the integral

a) Integral on a measure space

b) Monotone convergence theorem

c) Linearity of the integral

3. Constructions of measures

a) Abstract outer measure

b) Carathéodory theorem

c) Construction of the Lebesgue measure

4. Lebesgue integral

a) Lebesgue integral on the real line

b) Convergence theorems

c) Integrals depending on a parameter

5. Measure theory

a) Dynkin systems, uniqueness results

b) Premeasures, the Hopf theorem

c) Signed measures

d) Lebesgue decomposition and Radon-Nikodým theorem

e) Sequences of measurable functions, Jegorov theorem

f) Measurable mappings and push-forward of a measure

6. Multiple integrals

a) Product of measures, the Fubini theorem

b) Change of variables

c) Polar and spherical coordinates

7. L^p spaces

a) Basic definitions, equivalence classes

b) Hölder and Minkowski inequalities

c) Completeness

8. Lebesgue-Stieltjes integral

a) Regularity of measures

b) Lebesgue-Stieltjes measures and distribution functions

c) Integration by parts

d) Absolutely continuous and discrete cases