Description

Measure theory is the basis of modern theories of integration, of which Lebesgue integration is a typical case. The theory is quite general, permitting many notions of integration on a broad family of mathematical spaces. It is intimately tied to functional analysis, probability theory, and geometric analysis. This course will provide students with a broad introduction to the subject covering:

• General, Lebesgue and Hausdorff measures; Lusin’s and Egoroff’s theorems; Limit theorems for integrals; L^p spaces; Product measures and Fubini’s theorem; Radon-Nikodym and Riesz theorems; Area and coarea formulas; Sobolev spaces and capacities; BV functions; Sets of finite perimeter; Differentiability and approximation; Probability.

Prerequisites

The only assumed knowledge is the elementary real analysis with proofs. However, some prior exposure to integration, metric space and linear functional analysis would be a definite advantage.

Texts

• Measure theory and fine properties of functions, revised edition by Lawrence C. Evans and Ronald F. Cariepy, Chapman and Hall/CRC, 2015.
• Real analysis for graduate students, version 3.1 by Richard F. Bass, 2016.