Measure Theory - IITB

This is a course on the concepts of Measure and Integration. Normally, this is a core course for M,.Sc. Mathematics and Statistics students. The concepts find applications in advance Analysis Courses, Signal Processing, Financial Mathematics courses. INTENDED AUDIENCE : B.Tech Dual degree in Electrical, M.Sc. Physics, mathematicsPREREQUISITES : Basic Course in Real Analysis

Week 1   Lecture 1A Introduction, Extended Real Numbers Lecture 1B Introduction, Extended Real Numbers Lecture 2A Algebra and Sigma Algebra of Subsets of a Set Lecture 2B Algebra and Sigma Algebra of Subsets of a Set Lecture 3A Sigma Algebra generated by a Class Lecture 3B Sigma Algebra generated by a Class   Week 2   Lecture 4A Monotone Class Lecture 4B Monotone Class Lecture 5A Set Functions Lecture 5B Set Functions Lecture 6A The Length Function and its Properties Lecture 6B The Length Function and its Properties   Week 3   Lecture 7A Countably Additive Set Functions on Intervals Lecture 7B Countably Additive Set Functions on Intervals Lecture 8A Uniqueness Problem for Measure Lecture 8B Uniqueness Problem for Measure   Week 4   Lecture 9A Extension of Measure Lecture 9B Extension of Measure Lecture 10A Outer Measure and its Properties Lecture 10B Outer Measure and its Properties Lecture 11A Measurable Sets Lecture 11B Measurable Sets   Week 5   Lecture 12A Lebesgue Measure and its Properties Lecture 12B Lebesgue Measure and its Properties Lecture 13A Characterization of Lebesgue Measurable Sets Lecture 13B Characterization of Lebesgue Measurable Sets   Week 6   Lecture 14A Measurable Functions Lecture 14B Measurable Functions Lecture 15A Properties of Measurable Functions Lecture 15B Properties of Measurable Functions Lecture 16A Measurable Functions on Measure Spaces Lecture 16B Measurable Functions on Measure Spaces   Week 7   Lecture 17A Integral of Nonnegative Simple Measurable Functions Lecture 17B Integral of Nonnegative Simple Measurable Functions Lecture 18A Properties of Nonnegative Simple Measurable Functions Lecture 18B Properties of Nonnegative Simple Measurable Functions Lecture 19A Monotone Convergence Theorem and Fatou's Lemma Lecture 19B Monotone Convergence Theorem and Fatou's Lemma   Week 8   Lecture 20A Properties of Integrable Functions and Dominated Convergence Theorem Lecture 20B Properties of Integrable Functions and Dominated Convergence Theorem Lecture 21A Dominated Convergence Theorem and Applications Lecture 21B Dominated Convergence Theorem and Applications   Week 9   Lecture 22A Lebesgue Integral and its Properties Lecture 22B Lebesgue Integral and its Properties Lecture 23A Product Measure, an Introduction Lecture 23B Product Measure, an Introduction Lecture 24A Construction of Product Measures Lecture 24B Construction of Product Measures   Week 10   Lecture 25A Computation of Product Measure - I Lecture 25B Computation of Product Measure - I Lecture 26A Computation of Product Measure - II Lecture 26B Computation of Product Measure - II   Week 11   Lecture 27A Integration on Product Spaces Lecture 27B Integration on Product Spaces Lecture 28A Fubini's Theorems Lecture 28B Fubini's Theorems   Week 12   Lecture 29A Lebesgue Measure and Integral on R2 Lecture 29B Lebesgue Measure and Integral on R2 Lecture 30A Properties of Lebesgue Measure on R2 Lecture 30B Properties of Lebesgue Measure on R2 Lecture 31A Lebesgue Integral on R2 Lecture 31B Lebesgue Integral on R2