In the courses Analyse 1 and Analyse 2 you have become familiar with the Riemann integral on R and the Lebesgue integral. The Riemann integral, although natural and very important in applications, is not sufficient for several other areas in mathematics. The reason for this is that the definition is limited to functions on bounded domains, and that extensions to more general domains are described via limit transitions. The Riemann integral relates poorly to limits, e.g. interchanging integrals, differentiating under the integral, interchanging limit and integral for a sequence of functions, etcetera.
At the beginning of the 20th century Lebesgue developed a theory of integration, which was developend into an abstracted theory of integration based on general measure theory. Lebesgue's approach is an extension of the Riemann integral, which overcomes all the disadvantages mentioned above. After an introduction to the underlying structures of measure theory the corresponding integral theory is developed. Important theorems concerning convergence are discussed. This approach is important for other areas in mathematics, such as functional analysis (e.g. completeness of certain spaces of integrable functions), group theory (e.g. Haar measures on Lie groups) and probability theory. In the course we pay attention to applications in the latter area.
The goal of the course is to become familiar with the most important concepts in measure theory and the corresponding integration theory, in particular the convergence theory.
