[79] Intro to Lebesgue Measure & Lebesgue Integral (Baby Rudin Chapter 2 Set Theory #5) #4.3.2.2c5

Here is a small interlude to introduce Lebesgue Measure and the Lebesgue Integral. It's a companion to the last video on Riemann Integrability. Riemann integrals are limited in the kinds of functions that they can integrate. Lebesgue Integrals allow us to integrate a far larger class of functions. The machinery required to do so is complicated and fascinating and fun; that's what we go over in this video.

We are going to go through Chapter 2, "Basic Topology," of Rudin's Book, "Principles of Mathematical Analysis" (nicknamed "Baby Rudin"). This series will be the most challenging thing we have done yet! We are going to do a more thorough and careful introduction to topology than Rudin does, hopefully to better motivate the concepts he introduces in Chapter 2.

When we finish looking at topology, we will return to multivariable calculus, so we can acquire Green's Theorem, which is required for complex integration. Complex analysis is central to the proof of Fermat's Last Theorem.

With the first videos on this channel, we managed to keep the math very simple, almost exclusively nothing more advanced than algebra. Now, to complete our tour of elliptic curves, we have to take an ENORMOUS leap forward in the level of the mathematics involved.

I want to make sure we are all on the same page before we begin learning the more advanced stuff in earnest. I am going to assume we are all familiar with high school math and college math through first or second year calculus. I'm going to introduce complex numbers and complex integration in some detail after examining multivariable calculus.

This series will culminate with the Weierstrass Equation, which is the thing that connects elliptic curves to complex numbers, and thus allows us to connect them to modular forms, which is what Wiles's proof is all about.

Here is the outline of this series on complex numbers:
0. Introduction
1. The Real Number System
2. Multivariable Calculus (we are here)
3. Complex Numbers
4. Complex Functions
5. Exponential and Trigonometric Functions
6. Complex Integration
7. Cauchy's Integral Theorem
8. Cauchy's Integral Formula
9. Laurent Series
10. Complex Residues
11. Lattices and Doubly Periodic Functions
12. Lattices and Tori and Groups
13. The Weierstrass p-Function
14. The Weierstrass Equation: Complex Functions and Elliptic Curves

Please leave any questions, comments, or suggestions in the comments below!

Credits:
Music: "So Cruel" (cover of the U2 song) and "St. Jarna" by Depeche Mode