The field of real analysis focuses on mathematical analysis of real-valued functions of real variables, which critically depends on a robust notion of integration. Unfortunately, classic notions developed by Riemann and Darboux are limited to (piecewise) continuous functions, and are insufficient for a general theory. In this course we will generalize these notions using Lebesgue's theory, which provides a more delicate handling of integration by using a "total length" measure of subsets of the real line. We will explore the powerful properties of Lebesgue integration, as well as its relation to Riemann integration. We will then use Lebesgue integrals to consider function spaces and introduce a rigorous mathematical notion of "almost everywhere". Finally, one of the most important applications of Lebesgue's theory is in enabling the decomposition of periodic functions into a series of sines and cosines, which is known as Fourier series. This decomposition has tremendous significance both in theory and practice (e.g., signal processing), and we will spend a significant amount of time studying some of its properties & applications.
Homework assignments will be given on a weekly or bi-weekly basis, depending on progress in class. Skipping the submission of
The final grade will be based on homework assignments during the semester and a final exam. The exact ratio will be announced in class, but both parts will be significant.
The final exam will take place on Saturday, May 6th, 2:00 PM at WTS A60.
Here are a few note about the exam: