Introduction
Table of Contents
- 0. Overview of Measure Theory
- 1. Measures
- 1.1. \(\sigma\)-Algebras
- 1.2. Measures and Their Properties
- 1.3. Constructing Measures
Prerequisites
I’m assuming familiarity with metric space topology, the Riemann integral, and set theory. The first two are typically covered in undergraduate real analysis, and so I’ll be implicitly expecting familiarity with terminology like “infimum” and “supremum” as well. As for the set theory, I’m assuming familiarity with de Morgan’s laws and an understanding of infinite unions and intersections.
Other Remarks
I started learning measure theory during my last summer at UCLA. Some of my friends (including Mel) and I decided to patch our knowledge of real analysis because we just kind of skipped the graduate sequence for it. In our defense, our honours analysis professor (Prof. Greene) completely ravaged us, and we learned a lot from him. We even proved that \(S^1\) was not contractible, leading to intense recurrent nightmares about malformed pizza dough.
In our graduate complex analysis course, we had gotten used to spamming the dominated convergence theorem as a black box instead of really understanding where it comes from. We managed to get by, but we decided that it would be best to fill in that missing knowledge before moving forward.
We followed the homeworks from one of Prof. Terence Tao’s old real analysis courses. We covered the fall and winter quarters by following the readings and doing the homeworks.
Some of what I know about this comes from Prof. Tao’s own notes on real analysis, which you can find on his blog under Math 245A and Math 245B. Our main reference was Prof. Gerald Folland’s textbook on real analysis. A full list of references can be found below.
I want to thank Mel for being my study buddy and an overall great friend this summer. He’s given me a lot of motivation to learn on math, and he’s a big reason that I learned so much this summer.
References
Coming soon!