M4100 Mathematical Analysis IV

Faculty of Science
Spring 2025
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
In-person direct teaching
Teacher(s)
prof. RNDr. Roman Šimon Hilscher, DSc. (lecturer)
doc. Mgr. Peter Šepitka, Ph.D. (seminar tutor)
Guaranteed by
prof. RNDr. Roman Šimon Hilscher, DSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: 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 a 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 understand the abstract measure theory and the integral defined by a measure. In the special case one then gets the Lebesgue measure and the Lebesgue integral.
Learning outcomes
At the end of this course, the students will
- understand the Caratheodory construction of measurable sets and a measue,
- understand the construction of an abstract integral with respect to a measure,
- understand the construction of the Lebesgue measure and the Lebesgue integral,
- be able to explain the differences between the Lebesgue and Riemann integrals,
- understand the integration in product spaces,
- be able to analyze the behavior of functions defined as an integral depending on a parameter,
- be ready for applications of the measure theory and integration in differential equations, calculus of variations, and probability theory.
Syllabus
  • 1. Fundamental concepts of the measure theory: sigma-algebra, Borel set, measure, measurable sets.
  • 2. Outer measure and the Caratheodory construction of a measure.
  • 3. Lebesgue measure in Rn.
  • 4. Measurable functions.
  • 5. The abstract integral with respect to a measure, its basic properties, limit theorems.
  • 6. The Lebesgue integral in Rn, a comparison of the Lebesgue and Rieman integrals.
  • 7. The product of measures, integration in product spaces, the Tonelli and Fubini theorems.
  • 8. The substitution in the integral.
  • 9. Integrals depending on a parameter: continuity, differentiation, and their applications to the evaluation of definite integrals.
  • 10. Improper Lebesgue integral in Rn, Gamma and beta functions.
  • 11. L2 theory, Hilbert and Banach spaces.
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
  • LUKEŠ, Jaroslav and Jan MALÝ. Míra a integrál. 2. vyd. Praha: Karolinum, 2002, 179 s. ISBN 8024605430. info
  • NAGY, Jozef, Milan VACEK and Eva NOVÁKOVÁ. Lebesgueova míra a integrál. Vyd. 1. Praha: SNTL - Nakladatelství technické literatury, 1985, 151 s. URL info
Teaching methods
Two-hour lectures and two-hour exercises.
Assessment methods
Final exam has written and oral part (all in person). The results from exercises are partially transferred into the final grade (25% of the overall evaluations). The written and oral part of the exam contains also theoretical questions to proofs. The aim is to demonstrate the understanding of the basic concepts, their mutual relationship, and overall connections in the measure theory and integration. The conditions for final evaluation may be specified later depending on the pandemic situation and legal regulations.
Language of instruction
Czech
Further Comments
The course is taught annually.
The course is taught: every week.
The course is also listed under the following terms Spring 2021, Spring 2022, Spring 2023, Spring 2024.
  • Enrolment Statistics (Spring 2025, recent)
  • Permalink: https://is.muni.cz/course/sci/spring2025/M4100