M4170 Measure and Integral

Faculty of Science
Spring 2003
Extent and Intensity
2/2/0. 4 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Ladislav Adamec, CSc. (lecturer)
Guaranteed by
doc. RNDr. Bedřich Půža, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: doc. RNDr. Ladislav Adamec, CSc.
Timetable of Seminar Groups
M4170/01: No timetable has been entered into IS. L. Adamec
Prerequisites (in Czech)
M3100 Mathematical Analysis III
Diferenciální počet funkcí více proměnných. Metrické prostory.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The theory of measure and integration is nowadays a standard part of the basic course of mathematical analysis that is necessary for further successful study of modern mathematical analysis and its applications e.g. in the theory of differential equations or in the probability theory. In addition to the abstract theory of measure and abstract integration on measure spaces it contains the theory of Lebesgue integration in Rn and the integration of differential forms on k-dimensional submanifolds embedded in Rn.
Syllabus
  • 1) Fundamental concepts: Sigma-algebra, Borel set, measure, measurable sets
  • 2) Constructions of measures: Outer measures.
  • 3) Lebesgue measure in Rn: Outer Lebesgue measure in Rn, Lebesgue measurable sets.
  • 4) Measurable functions.
  • 5) The abstract Lebesgue integral.
  • 6) The Lebesgue integral in Rn.
  • 7) Fubini's theorem.
  • 8) Change of variable theorem.
  • 9) Integrals depending on a parameter.
  • 10) Differential forms and submanifolds n Rn.
  • 11) Surface and curve integrals.
  • 12) Integration of differential forms, integration on submanifolds in Rn, Stokes theorem.
Literature
  • KOLMOGOROV, Andrej Nikolajevič and Sergej Vasil‘jevič FOMIN. Základy teorie funkcí a funkcionální analýzy. Translated by Vladimír Doležal - Zdeněk Tichý. Vyd. 1. Praha: SNTL - Nakladatelství technické literatury, 1975, 581 s. info
  • RUDIN, Walter. Analýza v reálném a komplexním oboru. 1. vyd. Praha: Academia, 1977, 463 s. URL info
  • SIKORSKI, Roman. Diferenciální a integrální počet : funkce více proměnných. Translated by Ilja Černý. 2., změn. a dopl. vyd., Vyd. Praha: Academia, 1973, 495 s. URL info
Assessment methods (in Czech)
Podoba závěrečného hodnocení:písemná zkouška následovaná ústní zkouškou
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2011 - only for the accreditation, Spring 2000, Spring 2001, Spring 2002, Spring 2004, Spring 2005, Spring 2006, Spring 2007, Spring 2008, Spring 2009, Spring 2010, Spring 2011, Spring 2012, spring 2012 - acreditation, Spring 2013, Spring 2014, Spring 2015, Spring 2016, Spring 2017, spring 2018, Spring 2019, Spring 2020.
  • Enrolment Statistics (Spring 2003, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2003/M4170