Předmět Measure Theory and Integration (KMA / WTMAN)

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Materiál	Typ	Datum	Počet stažení

Obsah

1. Basic concepts from set theory and topology.2. Rings, sigma-rings, algebras and sigma-algebras of sets.3. Generated set systems, connection between some types of generated systems, examples of rings and sigma-rings over intervals.4. Concept of set function and some types of set functions. Properties of additive functions defined on rings of sets.5. Non-negative sigma-additive functions and their properties, measure, connection of measure with non-negative additive functions, examples of measures over intervals.6. Outer measure, measurability in sense of Caratheodory.7. Extension and completion of measure, m*-measurability and completeness, system of m*-measurable sets with respect to induced measure. Existence and uniqueness of extension of a measure.8. Lebesgue measure.9. Measurable space, simple measurable functions, measurable functions and criteria of measurability of functions.10. Further properties of measurable functions, sequences of measurable functions, Borel and Lebesgue measurability.11. Measure space, integral of simple functions.12. Integral of a non-negative measurable function, definition of integral of a measurable function and its properties.13. Integral and limit of a sequence of functions, integral as a set function, Lebesgue and Lebesgue-Stieltjes integral.

Získané způsobilosti

Studentgains basic knowledge of measure theory and integral, especially Lebesgue measure and integralanalyzes the theoretical arguments, understands the principles of evidence and application methods for solving the above-mentioned areas solves the specific tasks in this areaclassifies and synthesizes gained knowledge into the active formacquires global knowledge in this area

Literatura

JARNÍK, V. Integrální počet II. ACADEMIA Praha, 1976. NEUBRUNN, T., RIEČAN, B. Teória miery. VEDA Bratislava, 1992. Neubrunn, T., Riečan, B. Miera a integrál. VEDA, Vydavatelstvo Slovenskej Akadémie vied, Bratislava, 1981. LUKEŠ, J. Teorie míry a integrálu I. SPN Praha, 1972.

Požadavky

Written examination with possibility of oral supplement.The evaluation of the course including the classification is carried out in accordance with the Study and Examination Regulations OU.

Garant

doc. RNDr. Ladislav Mišík, CSc.

Vyučující

doc. RNDr. Ladislav Mišík, CSc.doc. RNDr. Ladislav Mišík, CSc.