Elements of Measure Theory

One of the most important tools which one combines with nonlinear analysis in the context of applied problems is “Measure Theory”. The subject started at the end of the nineteenth century with the works of Jordan, Borel, W.H. Young and Lebesgue. By that time it was clear to mathematicians that the Riemann integral had to be replaced by a new type of integral which will be more general (i.e. more functions will be integrable) and more flexible (in particular produce better convergence results). The construction of Lebesgue turned out to be the most fruitful and launched “Measure Theory” as a separate discipline in mathematical analysis. In contrast to the Riemann integral, the Lebesgue approach starts by partitioning the range of the function into small pieces, determining regions in the domain on which the function is approximately constant (these regions can be quite complicated) measuring the size of these regions, summing and passing to the limit as the size of the pieces in the range goes to zero. A prerequisite for this method to work, is the ability to measure the size of very general and complicated sets in the domain. This was the starting point of “Measure Theory”, which developed rigorously during the twentieth century. The aim of this chapter is to survey some parts of this theory which are needed in the understanding of certain aspects of nonlinear analysis. Of course our treatment is incomplete. Afterall this is impossible within a chapter of a book. We only present those items that are necessary for the discussion of future topics and special emphasis is placed on the interplay between Measure Theory and Topology.