Integration Theory
Integrationsteori

FMAP10, 7.5 credits, A (Second Cycle)

Valid for: 2025/26
Faculty: Faculty of Engineering LTH
Decided by: PLED F/Pi
Date of Decision: 2025-04-10
Effective: 2025-05-05

General Information

Depth of study relative to the degree requirements: Second cycle, in-depth level of the course cannot be classified
Elective for: F4, F4-mtm, Pi4-mtm
Language of instruction: The course will be given in English

Aim

The main goal of the course is to give a presentation of modern integration theory based on the general theory of measures. The 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 Lebesgue measure in R^d. These concepts are then used to define the integral of a measurable function 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.

Learning outcomes

Knowledge and understanding
For a passing grade the student must

• 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.

Competences and skills
For a passing grade the student must

• 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.

Judgement and approach
For a passing grade the student must

• 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.

Contents

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 in mean. Lp-spaces, Hölder's and Minkowski's inequalities,
• product measures, Fubini's and Tonelli's theorems.

Examination details

Grading scale: TH - (U, 3, 4, 5) - (Fail, Three, Four, Five)
Assessment:

The examination consists of a written examination and an oral examination at the end of the course. The oral examination may only be taken by those students who passed the written examination.

The final grade is based on the student's joint number of points on the two examinations. The written examination accounts for 75% of the total number of points and the oral one for 25%. For the grade 3 at least 50% of the total number of points are required, for the grade 4 at least 67% and for the grade 5 at least 80% of the total number of points.

The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.

Modules
Code: 0125. Name: Written Examination.
Credits: 5.0. Grading scale: UG - (U, G). Assessment: Written test comprising theory and problems. The module includes: The whole course.
Code: 0225. Name: Oral Examination.
Credits: 2.5. Grading scale: UG - (U, G). Assessment: Oral test on theory. The module includes: The whole course.

Admission

Admission requirements:

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

Reading list

• Cohn, Donald L: Measure Theory [Elektronisk resurs] Second Edition. New York, NY : Springer New York : Imprint: Birkhäuser, 2013, ISBN: 9781461469568.
• Stein, Elias M: Real Analysis [Elektronisk resurs]. 2010. Recommended complementary reading.

Contact

Teacher: Marcus Carlsson, marcus.carlsson@math.lu.se
Examinator: Anders Holst, Anders.Holst@math.lth.se
Director of studies: Anders Holst, Anders.Holst@math.lth.se
Director of studies: Anna-Maria Persson, anna-maria.persson@math.lu.se