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# Measure Theory - IMSc

IMSC and NPTEL via Swayam

## Overview

One of the main goals of Lebesgue's measure theory is to develop a fundamental tool for carrying out integration which behaves well with taking limits, and admitting a vast class of functions for which Riemann's integration theory is not applicable. Even though the crux of measure theory was to produce a good integration theory, it turns out that it also gives new ways of thinking about “measuring” objects, which is very useful for many other areas of mathematics such as probability theory as well as more advanced topics like harmonic analysis, ergodic theory, etc. Real-world applications of measure theory can be found in physics, economics, finance etc. Measure theoretic techniques are thus a must-have for any mathematician. INTENDED AUDIENCE :1st year M.Sc. onwardsPREREQUISITES :Set theory and Basic real analysis INDUSTRIES SUPPORT :None

## Syllabus

Week 1:Introduction and Motivation of Measure theory, Jordan measurability and Jordan content Week 2:Basic properties of Jordan content and connection with Riemann integrals, Motivation and definition of Lebesgue outer measure on R^n Week 3:Properties of Lebesgue outer measure on R^n, Caratheodory extension theorem Week 4:Lebesgue measurability, Vitali and Cantor sets, Boolean and sigma algebras
Week 5:Abstract measure spaces with examples: Borel and Radon measures, Metric outer measures, Lebesgue-Stieljes measures, Hausdorff measures and dimension* (extra content) Week 6:Measurable functions and abstract Lebesgue integration, Monotone convergence theorem, Fatou's lemma, Tonnelli's theorem Week 7:Borel-Cantelli Lemma, Dominated convergence theorem, the space L^1 Week 8:Various modes of convergence and their inter-dependence
Week 9:Riesz representation theorem, examples of measures constructed via RRT Week 10:Product measures and Fubini-Tonnelli theorem Week 11:Hardy-Littlewood Maximal inequality and Lebesgue's differentiation theorem Week 12:Lebesgue's differentiation theorem (continued)

### Taught by

Prof. Indrava Roy

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