Course abstract

One of the main goals of Lebesgue's measure theory is to develop a fundamental tool for carrying out integration which behaves well with taking limits, and admitting a vast class of functions for which Riemann's integration theory is not applicable. Even though the crux of measure theory was to produce a good integration theory, it turns out that it also gives new ways of thinking about “measuring” objects, which is very useful for many other areas of mathematics such as probability theory as well as more advanced topics like harmonic analysis, ergodic theory, etc. Real-world applications of measure theory can be found in physics, economics, finance etc. Measure theoretic techniques are thus a must-have for any mathematician.