Measure Theory - Period 1+2

We motivate and introduce the notion of a measure, that is, a way to assign a size to as many subsets as possible in an abstract space. It turns out that it is in general not possible to measure all sets, at least if one insists on countable additivity of the measure. This leads to the notion of a sigma-algebra. We show how one can obtain a unique measure on a sigma algebra once certain basic properties are imposed.

Once we have defined measure, we can introduce and discuss so called measurable functions which, roughly speaking, form the class of functions which we will be able to integrate. We then introduce and study integration of these measurable functions with respect to a measure. We discuss (among other things) the monotone and dominated convergence theorems concerning the interchangeability of limit and integral, the substitution rule, absolute continuity and the relation of this new integral to the Riemann integral. We also discuss multi-dimensional Lebesque measures, and Fubini's theorem.

The theory leads to a new perspective on integration of functions, which is not only more general than the Riemann setting when working on the real line, but also allows one to integrate in an abstract setting. This is of crucial importance for the development of functional analysis and probability theory.

Contact hours listed under a course description may vary due to the combination of lecture-based and independent work required for each course therefore, CEA's recommended credits are based on the ECTS credits assigned by VU Amsterdam. 1 ECTS equals 28 contact hours assigned by VU Amsterdam.