Measure And Integration

Overview

About the course:The theory of measure and integration is now an integral part of any masters or graduate program in mathemat ics in Indian Universities. It is an important prerequisite for most analysis based courses. In this course, we will start with a study of abstract measures with the Lebesgue measure being the most important example.After looking at measurable functions and some types of convergences for such functions, we will introduce the notion of the Lebesgue integral in an abstract measure space and study its properties. We will relate the two important processes of the calculus – differentiation and integration – in the context of the Lebesgue integral. We will also study measures and integrals on product spaces and also signed measures. Finally we will study the important properties of L^p – spaces.PRE-REQUISITES:MSc Real Analysis, Topology, Linear AlgebraINTENDED AUDIENCE: MSc (Mathematics) and above

Syllabus

Week 1: Motivation, abstract measures, Caratheodory’s method of extension, completion of a measure. Week 2:Construction of the Lebesgue meaure, approximation properties. Week 3:Translation invariance, nonmeasurable sets, measurable functions. Week 4:Properties of measurable functions, Cantor function. Week 5:Convergence, Egorov’s theorem, convergence in measure. Week 6:Lebesgue integration, convergence theorems. Week 7:Comparison with the Riemann integral, some applications (Weierstrass’ theorem). Week 8:Differentiation: Monotone functions, functions of bounded variation, absolute continuity. Week 9:Product spaces, Fubini’s theorem. Week 10:Signed measures, Radon-Nikodym theorem. Week 11:L^p-Spaces: Basic properties, approximation, applications. Week 12:Duality, convolutions.