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Theory of Real Functions 1 - NMMA403

Title:	Reálné funkce 1
Guaranteed by:	Department of Mathematical Analysis (32-KMA)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2016 to 2016
Semester:	winter
E-Credits:	4
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:	taught
Language:	Czech, English
Teaching methods:	full-time
Teaching methods:	full-time

Guarantor:	prof. RNDr. Luděk Zajíček, DrSc.
Class:	M Mgr. MA M Mgr. MA > Povinné
Classification:	Mathematics > Real and Complex Analysis
Incompatibility :	NRFA014
Interchangeability :	NRFA014
Is interchangeable with:	NRFA014

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Annotation -

Last update: T_KMA (10.05.2013)

Mandatory course for the master study branch Mathematical analysis. Recommended for the first year of master studies. Content: differentiation of measures, absolutely continuous functions, fuctions of bounded variation, Lipschitz function, Hausdorff measure and dimension.

Literature -

Last update: T_KMA (02.05.2013)

L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992.

W. Rudin, Real and Complex Analysis, Third edition. McGraw-Hill Book Co., New York, 1987.

Syllabus -

Last update: prof. RNDr. Ondřej Kalenda, Ph.D., DSc. (09.05.2022)

1. Differentiation of measures

covering theorems (Vitali, Besikovich, perhaps also Whitney)

maximal operator

absolutely continuous functions and functions of bounded variation (integration by parts, mean value theorems)

mutual differentiation of two Radon measures

Lebesgue points of locally integrable functions

approximate continuity and approximate differentiation

Rademacher theorem, relationship of Lipschitz functions and $W^{1,\infty}$

2. Hausdorff measure and dimension

outer Hausdorff measure

Hausdorff measure

Hausdorff dimension

connections to Lebesgue measure