Measure Theory

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)