Syllabus
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Course Code: MMATH21-407 Course Name: Elective IV) General Measure and Integration Theory |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Measures, some properties of measures, outer measures, extension of measures, uniqueness of extension, completion of a measure, the LUB of an increasingly directed family of measures.(Scope as in the Sections 3-6, 9-10 of Chapter 1 of the book ‘Measure and Integration’ by S.K. Berberian). Measurable spaces, measurable functions, combinations of measurable functions, limits of measurable functions, localization of measurability, simple functions (Scope as in Chapter 2 of the book ‘Measure and Integration’ by S.K. Berberian). |
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| 2 | Measure spaces, almost everywhere convergence, convergence in measure, almost uniform convergence, Egoroff’s theorem, Riesz-Weyl theorem (Scope as in Chapter 3 of the book ‘Measure and Integration’ by S.K. Berberian). Integrable simple functions, non-negative integrable functions, integrable functions, indefinite integrals, the monotone convergence theorem, mean convergence (Scope as in Chapter 4 of the book ‘Measure and Integration’ by S.K. Berberian) |
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| 3 | Product Measures: Rectangles, Cartesian product of two measurable spaces, sections, the product of two finite measure spaces, the product of any two measure spaces, product of two - finite measure spaces, Fubini’s theorem. (Scope as in Chapter 6 (except section 42) of the book ‘Measure and Integration’ by S.K. Berberian) Finite Signed Measures: Absolute continuity, finite singed measure, contractions of a finite signed measure, purely positive and purely negative sets, comparison of finite measures, Lebesgue decomposition theorem, a preliminary Radon-Nikodym theorem, Jordan-Hahn decomposition of a finite signed measure, domination of finite signed measures, the Radyon-Nikodym theorem for a finite measure space, the Radon-Nikodym theorem for a - finite measure space (Scope as in Chapter 7 (except Section 53) of the book ‘Measure and Integration’ by S.K.Berberian). |
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| 4 | Integration over locally compact spaces: continuous functions with compact support, G’s and F’s, Baire sets, Baire-sandwich theorem, Baire measures, Borel sets, Regularity of Baire measures, Regular Borel measures, Integration of continuous functions with compact support, Riesz-Markoff representation theorem (Scope as in relevant parts of the sections 54-57, 60, 62, 66 and 69 of Chapter 8 of the book ‘Measure and Integration’ by S.K.Berberian) | |
