Measure Theory
This course will enable the students to-
- Explore the concept of the abstract measure theory, definition and main properties of the integral.
- Construct Lebesgue's measure on the real line and in n-dimensional Euclidean space.
- Learn the advanced directions of measure theory.
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Course |
Learning outcomes (at course level) |
Learning and teaching strategies |
Assessment Strategies |
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Course Code |
Course Title |
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24DMAT 712 |
Measure Theory (Theory)
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CO163: Explore the basic concepts of set theory and measurability to define a measure, Non-measurable sets. CO164: Analyze the concept of measurable functions and convergence. CO165: Explore the bounded measurable functions with its properties. CO166: Diagnose square summable function. Create Parseval's identity, Riesz-Fischer theorem. CO167: Explore Lp-Spaces and the Hölder inequality, Minkowski inequalities. CO168: Contribute effectively in course-specific interaction.
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Approach in teaching: Interactive Lectures, Discussion, Power Point Presentations, Informative videos
Learning activities for the students: Self learning assignments, Effective questions, presentations, Assigned tasks |
Class Test, Individual and group projects, Open Book Test, Semester End Examination
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Algebra and algebras of sets, Algebras generated by a class of subsets, Borel sets, Lebesgue measure of sets of real numbers, Measurability and measure of a set, Existence of non-measurable sets.
Realization of non-negative measurable function as limit of an increasing sequence of simple functions, Structure of measurable functions, Convergence in measure, Egoroff's theorem.
Lebesgue integral of bounded measurable functions, Lebesgue theorem on the passage to the limit under the integral sign for bounded measurable functions.
Space of square summable functions, Fourier series and coefficients, Parseval's identity, Riesz-Fisher Theorem.
Lp-spaces, Holder-Minkowski inequalities, Completeness of Lp-spaces.
- Shanti Narayan, A Course of Mathematical Analysis, S.Chand & Co.New Delhi, 2005.
- T.M. Apostol, Mathematical Analysis, Narosa Publishing House, New Delhi, 2002.
- Walter Rudin, Real and Complex Analysis, McGraw-Hill Education, 2017.
e- RESOURCES
● https://nptel.ac.in/courses/111108135
● https://nptel.ac.in/courses/111101100
● https://nptel.ac.in/courses/111106140
● https://onlinecourses.nptel.ac.in/noc21_ma21/preview
JOURNALS
● https://terrytao.files.wordpress.com/2012/12/gsm-126-tao5-measure-book.pdf
● https://library.oapen.org/bitstream/id/ce19d94d-b8b6-420f-9e69-d9f565703c26/1007045.pdf
● https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes.pdf
- P.K. Jain and S.K. Kaushik, An Introduction to Real Analysis, S.Chand & Co, New Delhi, 2000.
- R.R. Goldberg, Real Analysis, Oxford and IBH publishing Company, New Delhi, 1970.
- Halsey Royden, Patrick Fitzpatrick, Real Analysis, Pearson’s United States Edition, 2010.
- G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill Book, New Delhi, 2017.
- G. De. Barra, Measure Theory and Integration, New Age International Private Limited, 2013.
- S.K. Berberian, Measure and Integration, McMillan, New York, 1965.
- I.K. Rana, An Introduction to Measure and Integration, Narosa Publishing House New Delhi, 2007
- https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes.pdf
