Description
Lebesgue's theory of measures and integration is one of the most celebrated achievement of modern Analysis in the 20th century. Lebesgue's theory not only resolved many of the issues we had with the Riemann integral on \(\mathbb{R}^n\), but its notion of measure and integration is at the core of our understanding of fundamental functional properties and function spaces, as well as Probability Theory. The goal of our project is to explore various topics in Lebesgue's theory - focusing mostly on \(\mathbb{R}\) and \(\mathbb{R}^n\).
We will start the project by exploring how Lebesgue's theory helps us understand not only integration - but also differentiation. We will cover topics such as
- The Vitali covering theorem and the Hardy-Littlewood maximal function.
- Lebesgue's differentiation theorem.
- The Lebesgue set of a function.
- Functions of bounded variation.
- Absolute Continuity and the Fundamental Theorem of Calculus.
- Rademacher's theorem on \( \mathbb{R} \).
- Lebesgue integration on \(\mathbb{R}^n\) and Fubini's theorem.
- Change of variables.
- Nowhere differentiable functions.
- Convex Functions.
- Rademacher's theorem on \( \mathbb{R}^n\).
- General measure theory.
- The Radon-Nikodym theorem.
- Weak derivatives and Sobolev spaces.
Prerequisites
Prerequisites (essential) : Analysis III.
Resources
- Jones F. Lebesgue Integration on Euclidean Space.
- Royden, H. L. and Fitzpatrick, P. Real Analysis.