English

Every spring semester

The main goal of the course is to give a presentation of modern integration theory based on the general theory of measures. The PhD students will aquire a powerful and general machinery applicable to important problems in analysis as well as in other areas of mathematics, especially probability theory, partial differential equations and spectral theory.
This includes the general notion of a measure defined on a sigma-algebra, construction of measures with help of outer measures, in particular the Lebesguemeasure in R^d. These concepts are then used to define the integral of a measurablefunction with respect to a given measure and study its properties. The focus is on convergence theorems, that is, interchanging limits and integrals, as well as multiple integrals which appear as integrals against measures on product spaces.

The course treats the definition and fundamental properties of measures and integrals on general measurable spaces:
*definition of measures and construction with help of outer measures.
*the Lebesgue measure on R^d and Lebesgue-Stieltjes measures on the real line,
*measurable functions and their integrals with respect to a given measure.
*the Lebesgue integral on R^d and its comparison with the Riemann integral,
*the monotone and dominated convergence theorems, Fatou's lemma,
*pointwise almost everywhere convergence, convergence in measure and inmean. Lp-spaces, Hölder's and Minkowski's inequalities,
*product measures, Fubini's and Tonelli's theorems.

be able to give a detailed account of the concepts, theorems and methods within integration theory that are treated in the course,

be able to identify the main theorems of the course, describe the main ideas and carry out the steps in their proofs,

be able to give examples of non-trivial situations where these theorems apply,

be able to give a detailed account of the relation between the Riemann and Lebesgue integral of a function defined on a compact interval.

be able to integrate knowledge from the different parts of the course in connection with problem solving,

be able to identify problems that can be solved by methods that are part of the course and solve these using appropriate solution methods,

be able to explain the solution to related mathematical problems, in speech and in writing, logically coherent and with adequate terminology.

be able to identify situations where the methods of integration theory apply, for example in other areas such as probability theory, partial differential equations, function spaces.

Lectures

Seminars

Written exam

Oral exam

Failed, pass

(FMAB30 Calculus in Several Variables or FMAB35 Calculus in Several Variables) and (FMAF01 Mathematics - Analytic Functions or FMAA60 Introduction to Real Analysis)

Cohn, Donald L.: Measure Theory: Second Edition. Springer Science & Business Media, 2013. ISBN 9781461469568.

FMAP10F

2025-01-10

Maria Sandsten