M4170 Measure and Integral

Faculty of Science
Spring 2011 - only for the accreditation
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 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
Prerequisites
M3100 Mathematical Analysis III
Differential and integral calculus in several veriables, metric spaces.
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.
The aim of the course is to give a slow introduction to the measure theory as well as to the theory of Lebesgue integration.
At the end of this course, students will acquire working knowledge of the field.
Students will be able to use the theory of abstract measure, the theory of abstract Lebesgue integration on measure spaces and the theory of Lebesgue integration in Rn in modern mathematical analysis and its applications e.g. in the theory of differential equations or in the probability theory.
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. Comparision of Lebesgue and Rieman integrals.
  • 7) Fubini's theorem.
  • 8) Parametric integrals.
  • 9) Change of variable theorem.
Literature
  • RUDIN, Walter. Analýza v reálném a komplexním oboru. Vyd. 2., přeprac. Praha: Academia, 2003, 460 s. ISBN 8020011250. info
  • 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
  • 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
Teaching methods
Lectures, class exercises.
Assessment methods
Written examination followed by an oral examination.
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught annually.
The course is taught: every week.
The course is also listed under the following terms Spring 2008 - for the purpose of the accreditation, Spring 2000, Spring 2001, Spring 2002, Spring 2003, 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.