In a few weeks (and more precisely, starting Friday, September 24), I will begin teaching Math 245A, which is an introductory first year graduate course in real analysis. (A few years ago, I taught the followup courses to this course, 245B and 245C.)  The material will focus primarily on the foundations of measure theory and integration theory, which are used throughout analysis.  In particular, we will cover

1. Abstract theory of $\sigma$-algebras, measure spaces, measures, and integrals;
2. Construction of Lebesgue measure and the Lebesgue integral, and connections with the classical Riemann integral;
3. The fundamental convergence theorems of the Lebesgue integral (which are a large part of the reason why we bother moving from the Riemann integral to the Lebesgue integral in the first place): Fatou’s lemma, monotone convergence theorem, and the dominated convergence theorem;
4. Product measures and the Fubini-Tonelli theorem;
5. The Lebesgue differentiation theoremabsolute continuity, and the fundamental theorem of calculus for the Lebesgue integral.  (The closely related topic of the LebesgueRadon-Nikodym theorem is likely to be deferred to the next quarter.)

See also this preliminary 245B post for a summary of the material to be covered in 245A.

Some of this material will overlap with that seen in an advanced undergraduate real analysis class, and indeed we will be revisiting some of this undergraduate material in this class.  However, the emphasis in this graduate-level class will not only be on the rigorous proofs and on the mathematical intuition, but also on the bigger picture.  For instance, measure theory is not only a suitable foundation for rigorously quantifying concepts such as the area of a two-dimensional body, or the volume of a three-dimensional one, but also for defining the probability of an event, or the portion of a manifold (or even a fractal) that is occupied by a subset, the amount of mass contained inside a domain, and so forth.  Also, there will be more emphasis on the subtleties involved when dealing with such objects as unbounded sets or functions, discontinuities, or sequences of functions that converge in one sense but not another.  Being able to handle these sorts of subtleties correctly is important in many applications of analysis, for instance to partial differential equations in which the functions one is working with are not always a priori guaranteed to be “nice”.