Real Analysis II
Fall 2015- The Humke Version
Real Goals for Real II
-- Fall 2015
- The Prime Directive
Hey, the Prime Directive is twofold -- have a little fun and learn a little
stuff. And I'm dead serious about this!! If we don't accomplish
both of these I won't be satisfied with the term and I imagine you
won't be either. That said, this course is really mission critical for those of you with
grad school in stat, math or mathematical econ in your future. So we have a few
topics we should really go over to help you lay a strong foundation.
- What IS This Anyway?
This course is about the theory of Lebesgue measure and integration. We'll begin by examining some defects of the Riemann integral and then develop Lebesgue's ideas of how to circumvent these defects. Although Lebesgue
did not complete the task, his theory forms the basis for the modern theory of integration and an inherent measure theory.
But this "measure theory" is not just probabilistic in nature, it IS the language of modern probability theory as it is understood and practiced. Often during the semester, I'll use probability as motivation for definitions and proofs.
I'll also hold several evening sessions for those who are particularly interested in statistics. Statisticians and probablists use slightly different terminology and have a slightly different viewpoint than the measure theory folks. During those evening sessions I'll drop the gloves and go probablistic on you.
Capinski and Kopp follow an historical line of thought which, although not the most direct route, reveals much about how the theory of integration is closely tied to the geometry of functions and the foundations of probability. Topics covered include:
- Definition of measure and set examples
- Sigma fields Forever
- Definition of the Lebesgue Integral and Basic Results
- Measurable Sets and Functions
- Convergence Theorems.
Only 5 items you say? It'll be enough!