Studying and doing mathematics:
Some of our readings and assignments on measure theory will come from parts of
Mathematical Analysis by T. Apostol;
An Introduction to Measure Theory by T. Tao;
Principles of Mathematical Analysis by W. Rudin;
An article about the Cantor set, which we will see on a few occasions.
Here is the construction of a Borel set that is not measurable, showing that the Borel algebra is strictly smaller than the algebra of Lebesgue measurable sets.
We briefly mentioned Littlewood's Three Principles in class. Here are the precise statements and proofs. You can also read more about them in Tao's book linked above.
Lebesgue’s presentation of the construction of the integral from 1926.
limsup and liminf will make an appearance in a few crucial places throughout the semester. Here is the Wikipedia page on these concepts. Here and here and discussions of liminf and limsup of sequences of numbers, and here is something that explains how to think of the liminf of a sequence of functions.
Here is a page that gives a couple of definitions of expected value (one uses the integral of a random variable and one uses the integral of its probability density function) and also talks about the Radon-Nikodym Theorem.
Measure theory has many applications. Some probability and Fourier analysis are covered in this class, but we could have also talked about applications in economics, finance, physics, etc. Here is a nice survey article about some of the ways measure theory is used in economics. If you are interested in further materials, please feel free to contact me.
If you're interested in reading further or filling in the details of the parts of functional analysis we covered, here is a nice document that covers all the basics.
For more on applications of Fourier analysis, look at the first chapter of this book. Here is another good resource. And one more. And this is a page where an animation of the successive Fourier approximations to a function is shown. Here is another page with nice images. And here is a nice overview of the usefulness of Fourier series, written by our own Prof. Schultz. Here is an applet that lets you see and hear(!) some Fourier approximations. Finally, here's a video of how Fourier analysis makes a piano speak!