(guy.wolf@yale.edu)

This course is concerned with developing an abstract mathematical notion for the intuitive (and rather common) operation of "measuring things". This notion will be used to generalize concepts of integration (or summation) from Calculus, which are classically defined using neighborhoods of infinitesimal length (or volume in the multivariate case).

Interestingly, as we will see in the first class, even simple intuitive everyday quantities we expect to be able to measure, such as length or volume, cannot be consistently defined for all subsets of the Euclidean space (see, for example, the Banach-Tarski paradox). Therefore, this course will start by defining and exploring the properties of sigma-algebras, measures, and outer measures, in order to provide a formal mathematical interpretation for how to "measure things" and what "things" can be measured to begin with, thus also providing formal meaning to measurable sets and functions.

Following this foundation, we will introduce the Lebesgue integral, which is a powerful generalization of the more elementary notions of antiderivatives and Riemann sums, and discuss some of its properties. We will then move on to consider ways of comparing measures and study the Radon-Nikodym theorem. Using some notions of point set topology, which will be provided at an introductory level during the course, we will try to understand the full power of the Lebesgue integral by studying the beautiful structures that arise when we look at function spaces. This will also lead us to basic notions in functional analysis and Lebesgue spaces. If time allows, we may also survey some additional notions related to integration and measure spaces.

Required: Real Analysis for Graduate Students, Version 3.1, Richard F. Bass, 2016.Recommended: Real Analysis: Modern Techniques and Their Applications, 2nd Edition, Gerald B. Folland, 1999.

- Exercise 1 (optional) - due 09/12/2017 2:30 PM
- Exercise 2 - due 09/19/2017 2:30 PM
- Exercise 3 - due 09/26/2017 2:30 PM
- Exercise 4 - due 10/03/2017 2:30 PM
- Exercise 5 - due 10/10/2017 2:30 PM
- Exercise 6 - due 10/17/2017 2:30 PM
- Exercise 7 - due 10/26/2017 2:30 PM
- Exercise 8 - due 11/02/2017 2:30 PM
- Exercise 9 - due 11/09/2017 2:30 PM
- Exercise 10 - due 11/16/2017 2:30 PM
- Exercise 11 - due 11/30/2017 2:30 PM
- Exercise 12 - due 12/08/2017 5:00 PM

The final exam will take place on **Sunday, Dec. 17th, 7:00 PM**

Here are a few note about the exam:

- You will have 150 minutes for solving the problems and additional 30 minutes for reviewing your answers.
- I recommend that you arrive at least 15 minutes before the exam starts.
- Please try to make your restroom break prior to the exam, so you will not have to go during it. In any case, the exam will end on time even if you are stuck out of class, and 0 points will be given if no solution is submitted on time (barring official dean's letters).
- You can bring with you one handwritten double-sided page of A4 or Letter size. Please use reasonable size for your handwritten notes on it, since a magnifying glass is not included in the allowed equipment for the test.
- As you can imagine, any discovered attempt at cheating will result in 0 points on the exam.