Mathematics: Integration Theory
Gain a rigorous understanding of measure and integration theory—the foundation of modern analysis. This course introduces key concepts such as sigma-algebras, measures, and Lebesgue integration, providing essential tools for advanced studies in analysis, probability theory, and applied mathematics.
Start
Autumn 2026
Level
Master's
Language
English
Place of study
Lund
Course code
MATM39
This course presents modern integration theory through the framework of measure theory. You will learn how to define and work with measures on abstract spaces and apply these concepts to develop powerful methods for solving problems in analysis and related fields. The course provides a solid foundation for advanced studies in analysis, probability theory, and applied mathematics.
Course content includes
- Measures on sigma-algebras
- Construction of measures using outer measures
- Lebesgue measure on ℝⁿ
- Measurable functions and their integrals
- Comparison between Riemann and Lebesgue integrals
- Convergence theorems (monotone, dominated, Fatou’s lemma)
- Pointwise and mean convergence
- Product measures and repeated integration
- Lᵖ spaces, Hölder’s and Minkowski’s inequalities
- Fubini’s and Tonelli’s theorems
The course is alternative-compulsory within the Master’s Programme in Mathematics and Numerical Analysis at Lund University. It is also available as a standalone course, and can be taken as an elective course by Bachelor’s students in mathematics.
The course combines lectures and seminars where theory meets practice. You will work through rigorous proofs, solve problems, and discuss key concepts in small groups.
Assessment consists of a written exam (5 credits) and an oral exam (2.5 credits). The oral exam is only available to students who pass the written exam. Re-examinations are offered shortly after the regular ones.
Integration theory is a cornerstone of modern mathematics. After this course, you will be well-prepared for advanced studies in areas such as probability theory, functional analysis, and partial differential equations.
The skills you gain - rigorous reasoning, abstract thinking, and problem-solving - are highly valued in research, data science, finance, and other analytical professions.
Entry requirements
For admission to the course English 6 or the equivalent, and at least 90 higher education credits in science or engineering, including at least 60 credits in mathematics, are required.
Selection criteria
Seats are allocated according to: ECTS (HPAV): 100 %.
Tuition fees for non-EU/EEA citizens
Citizens of countries outside:
- The European Union (EU)
- The European Economic Area (EEA) and
- Switzerland
are required to pay tuition fees. You pay an instalment of the tuition fee in advance of each
semester.
Tuition fees, payments and exemptions
Full programme/course tuition fee: SEK 23,125
First payment: SEK 23,125
Note that you may also need to pay an application fee, or provide proof of exemption.
No tuition fees for citizens of the EU, EEA and Switzerland
There are no tuition fees for citizens of the European Union (EU), the European Economic Area (EEA) and Switzerland.
A shared second home for mathematics
Matematikcentrum gathers students and researchers in mathematics, mathematical statistics, and numerical analysis and computational science. Along Sölvegatan, close to green spaces and the wider university area, you find a shared environment for everyone who uses mathematics as a common language. Research here has led to innovations and companies in areas such as optimisation, image analysis and software, with applications ranging from logistics to video analysis and mobile technology.
