Measure theory course notes

Here are my class notes for the measure theory course offered by Prof B V Rao in August-November 2010.

• Lecture 1, 2nd August (intervals, semi-fields, measures, extending measures to the field)
• Lecture 2, 4th August (arithmetic with infinities, towards the countable additivity for the semi-field of intervals)
• Lecture 3, 6th August (countable additivity for the semi-field of intervals, σ-fields)
• Lecture 4, 9th August (more on σ-fields, measures, continuity of measures, Carathéodory extension theorem, reducing it to the finite measure case)
• Lecture 5, 11th August (proof of the Carathéodory extension theorem)
• Lecture 6, 30th August (proof of the Carathéodory extension theorem, monotone classes)
• Lecture 7, 1st September (proof of the monotone class theorem, an application to uniqueness of extensions, motivation behind Lebesgue integration)
• Lecture 8, 3rd September (measurable functions and their properties, simple functions)
• Lecture 9, 8th September (approximation of measurable functions by simple functions, recognising all measurable functions, bounded pointwise convergence, overview of defining the Lebesgue integral)
• Lecture 10, 13th September (uniform convergence vs bp convergence, defining integration, basic properties, monotone convergence theorem)
• Lecture 11, 15th September (defining integration, basic properties, Fatou's lemma, dominated convergence theorem)
• Lecture 12, 17th September (defining integration, examples (summation of series and Riemann integration), more properties of integration)
• Lecture 13, 20th September (more properties of integration, product of finite measure spaces, statement of Fubini's theorem for nonnegative functions)
• Lecture 14, 22nd September (product of finite measure spaces (contd), the need for allowing functions to take the value ∞)
• Lecture 15, 24th September (extended real measurable functions)
• Lecture 16, 4th October (extended real measurable functions and integration, the art of handling null sets)
• Lecture 17, 6th October (Fubini's theorem for nonnegative functions and integrable functions)
• Lecture 18, 8th October (Fubini's theorem for integrable functions (contd), products of sigma-finite measure spaces)
• Lecture 19, 11th October (Fubini's theorem for sigma-finite measure spaces, application of Fubini's theorem - integration is area under the curve, products of finitely many measure spaces, Radon-Nikodym theorem)
• Lecture 20, 13th October (Proof of the Radon-Nikodym theorem)
• Lecture 21, 15th October (Absolute continuity and singularity, function spaces)
• Lecture 22, 18th October (More on convergence of functions)
• Lecture 23, 20th October (L^p is a vector space, statements of Hölder's and Minkowski's inequalities)
• Lecture 24, 22nd October (Convex functions, proofs of Hölder's and Minkowski's inequalities)
• Lecture 25, 25th October (Integration of complex valued functions, complex L^p spaces)
• Lecture 26, 27th October (Differentiating under the integral sign, completeness of L^p)
• Lecture 27, 29th October (More on L^p, Jensen's inequality)
• Lecture 28, 1st November (Jensen's inequality (contd), Lebesgue decomposition theorem)
• Lecture 29, 10th November (Products of countably many probability spaces on R)
• Lecture 30, 12th November (Products of countably many probability spaces on R (contd))
• Lecture 31, 15th November (Haar measure (introduction))
• Lecture 32, 19th November (Haar measure (contd), ultrafilters and convergence)
• Lecture 33, 22nd November (Construction of Haar measure)
• Lecture 34, 24th November (Construction of Haar measure (contd))
• Lecture 35, 26th November (Construction of Haar measure (contd), uniqueness)
• All lectures in one file. This was obtained by concatenating the pages of the above files, so the page numbers shown are as in the original files.

Here are the homework sets 1 to 13.