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Measure and Integration Theory - NMAA068

Title:	Teorie míry a integrálu
Guaranteed by:	Department of Mathematical Analysis (32-KMA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2011
Semester:	winter
E-Credits:	9
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:	cancelled
Language:	Czech
Teaching methods:	full-time
Teaching methods:	full-time

Class:	Předměty bloku A
Classification:	Mathematics > Real and Complex Analysis
Interchangeability :	NMAA069, NMAA070

Opinion survey results Examination dates Schedule Noticeboard

Annotation -

Last update: T_KMA (20.05.2004)

Elements of the measure theory, background for probability theory. Abstract integration on measure spaces. Introduction of the Lebesgue measure. Calculus of integrals over domains, curves and surfaces. Replaced by MAA069 and MAA070

Literature - Czech

Last update: RNDr. Pavel Zakouřil, Ph.D. (05.08.2002)

B. P. Děmidovič: Sbornik zadač i upražněnij po matematičeskomu analizu

J. Kopáček: Matematika pro fyziky IV, skripta MFF

J. Lukeš: Příklady z matematické analýzy I. Příklady k teorii Lebesgueova integrálu, skripta MFF

J. Lukeš: Teorie míry a integrálu I, skripta MFF

J. Lukeš, J. Malý: Míra a integrál (Measure and integral), skripta

I. Netuka, J. Veselý: Příklady z matematické analýzy. Míra a integrál, skripta

W. Rudin: Analýza v reálném a komplexním oboru

W. Rudin: Základy analýzy v reálném a komplexním oboru

Syllabus -

Last update: T_KMA (22.05.2003)

1. Foundations of measure theory.

Sigma - algebra, Borel sets, measure, complete measure, measurable functions, simple functions.

2. Lebesgue measure in Rn.

Outer Lebesgue measure and measurable sets. Lebesgue measure and its properties.

3. Abstract integral.

Construction of integral on a measure space. Fatou lemma, Levi a Lebesgue theorems (monotone convergence, dominated convergence). Chebyshev's inequality nerovnost.

4. Integral and measure in R.

Relation of Lebesgue, Newton and Riemann integral. Distribution functions and Lebesgue-Stieltjes measure.

5. Integral depending on a parameter.

Continuity, differentiation. Applications in calculus, Gamma function and Beta function.

6. Integral calculus in Rn.

Fubini theorem in Rn, change of variables, polar, spherical and cylindrical coordinates. Laplace integral.

7. Lp spaces and convergence of sequences of functions.

Almost everywhere convergence, Jegorov theorem.

8. Measure theory.

Product of measures, abstract Fubini theorem. Push forward of a measure. Radon - Nikodym theorem and Lebesgue decomposition. Signed measures. Hahn and Jordan decomposition.

9. (optional.)

Curve and surface integration. Potential of a vector field. Divergence theorem, Stokes theorem.