• is familiar with sigma-algebras, positive and complex measures
• understands various constructions of measures (Caratheodory, Lebesgue) and Riesz' representation theorem
• knows product measures and Fubini's theorem
• understands the fundamental theorem of calculus in the Lebesgue setting
• has seen some constructions relevant for probability theorem, like conditional expectations

In Analysis 1+2 you have encountered Riemann integration of functions on bounded regions in R and Rⁿ. The definition of the Riemann integral and the proof of Riemann integrability for monotonous (for n=1) or continuous functions are quite simple. But the Riemann integral has many defects:

1. The definition applies only to bounded functions on bounded regions; in more general situations, one needs to take limits (improper Riemann integral)
2. But even under these restrictions there are many functions that are not integrable, for example the characteristic function of the set of rational numbers
3. Even more seriously, there are few useful results on the interchange of Riemann integration with limits or about changing the order of two integrations

For functions on Rⁿ these problems are solved by the Lebesgue integral that has been developed by Lebesgue around 1900. But Lebesgue's integration theory can be generalized further to all spaces equipped with a `positive measure'. Such a space can be `infinite dimensional' or have no topology. This generality is crucial for many applications of integration theory, for example to probability theory and statistical mechanics, to functional analysis, group theory, etc.

The aim of this course is to provide the necessary knowledge and abilities needed for the deeper study of subjects like those just mentioned.