Syllabus
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Course Code: MMATH20-204 Course Name: Measure and Integration |
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| MODULE NO / UNIT | COURSE SYLLABUS CONTENTS OF MODULE | NOTES |
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| 1 | Lebesgue outer measure, elementary properties of outer measure, measurable sets and their properties, Lebesgue measure of sets of real numbers, algebra of measurable sets, Borel sets and their measurability, characterization of measurable sets in terms of open, closed, F and G sets, existence of a non-measurable set. |
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| 2 | Lebesgue measurable functions and their properties, the almost everywhere concept, characteristic functions, simple functions, approximation of measurable functions by sequences of simple functions, Borel measurability of a function.
Littlewood’s three principles, measurable functions as nearly continuous functions. Lusin’s theorem, almost uniform convergence, Egoroff’s theorem, convergence in measure, F.Riesz theorem that every sequence which is convergent in measure has an almost everywhere convergent subsequence. |
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| 3 | The Lebesgue Integral: Shortcomings of Riemann integral, Lebesgue integral of a bounded
function over a set of finite measure and its properties, Lebsegue integral as a generalization of the Riemann integral, Bounded convergence theorem, Lebesgue theorem regarding points of discontinuities of Riemann integrable functions. Integral of a non-negative function, Fatou’s lemma, Monotone convergence theorem, integration of series, the general Lebesgue integral, Lebesgue convergence theorem. |
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| 4 | Differentiation and Integration: Differentiation of monotone functions, Vitali’s covering lemma, the four Dini derivatives, Lebesgue differentiation theorem, functions of bounded variation and their representation as difference of monotone functions. Differentiation of an integral, absolutely continuous functions and their properties, convex functions, Jensen’s inequality. The Lp-spaces and their completeness. |
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