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NPTEL

Measure Theory

NPTEL via Swayam

Overview

This course covers measure and integration. We start with abstract measures and their integration theory. Next, we construct the Lebesgue measure and follow it with a detailed study of Borel measures on locally compact Hausdorff spaces. Lp spaces and product measures along with Fubini’s theorem is taken up next. We finish with several classical reasul, Radon-Nikodym theorem, Ries representation theorem and Lebesgue differentiation theorem.     INTENDED AUDIENCE : First year MSc students in mathematics PREREQUISITES : A course in real analysis and topology INDUSTRY SUPPORT : Nil

Syllabus

COURSE LAYOUT Week 1 : Abstract measures and integration (3 lectures) Week 2 : Abstract measures and integration (3 lectures) Week 3 : Outer measure on Rn and properties (3 lectures) Week 4 : Lebesgue measure and integration (3 lectures) Week 5 : Borel measures on locally compact spaces (3 lectures) Week 6 : Lp – spaces and properties (3 lectures) Week 7 : Product measures (2 lectures) Week 8 : Product measures (2 lectures) Week 9 : Complex measures and Radon-Nikodym theorem (2 lectures) Week 10 : Dual of Lp –spaces (2 lectures) Week 11 : Riesz representation theorem (2 lectures) Week 12 : Lebesgue differentiation theorem and absolutely continuous functions (2 lectures)

Taught by

Prof. E. K. Narayanan

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