Course Details
Subject {L-T-P / C} : MA4102 : Measure Theory { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Prof. Divya Singh
Syllabus
Lebesgue outer measure, Measurable sets and Lebesgue measure, Non-measurable sets, Semi-algebras, algebras, monotone class, sigma-algebras, Borel sets, Measurable functions, Simple functions, Littlewood’s three principles, Lebesgue integral, Fatou’s lemma, Lebesgue monotone convergence theorem, Lebesgue dominated convergence theorem, Lp spaces, Minkowski's and Holder's inequalities, Completeness of Lp spaces, Convergence in measure, Differentiation of monotone functions, Functions of bounded variation, Absolute continuity, Abstract measure spaces, Completion of measure, Product measure, Fubini’s theorem.
Course Objectives
- To introduce the concept of Lebesgue integral via Lebesgue measure on real line, and extend this idea to abstract spaces and higher dimensional Euclidean spaces.
Course Outcomes
1. Students will be able to understand the limitations of Riemann integration and the need of extension of the concept of length/volume to measure. <br />2. They will get acquainted with the Lebesgue integral and related results. <br />3. They will be able to do manipulations of sums and integral.
Essential Reading
- H. L. Royden, Real Analysis, Macmillan Publishing Company
- G. De Barra, Measure Theory and Integration, Ellis Horwood Publishing Corporation
Supplementary Reading
- E. D. Benedetto, Real Analysis: Foundations and Applications, Springer
- I. K. Rana, An Introduction to Measure and Integration, Narosa Publishing House