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The idea of measuring physical quantities (like length/area/volume etc) of certain geometrical objects is well-studied using Riemann integration. However, there are objects whose length/area/volume could not be determined via Riemann integration. Hence, this method needs to be further strengthened. Lebesgue (and others) had taken up the issue of further extending the idea of measuring certain physical quantities related to rough (highly discontinuous) and unbounded objects by decomposing an object into countably many elementary small pieces with an intuition of recollecting them without any quantitative alteration. Those elementary pieces are nearly open (or closed) sets and are known as measurable sets.

However, Lebesgue's ideas could not have succeeded in determining the area of certain objects, which are rough beyond expectation. Those objects are considered as non-measurable sets. On the other hand, if the given object is a sector bounded by curves, then one can compute its area via Riemann integration, while, these curves are almost continuous. If the boundary of an object cannot be decomposed onto the images of finitely many almost continuous functions, then one needs to further generalize the idea of ''almost continuous'' function. Those almost continuous functions are generalized as measurable functions, which are nearly continuous and the method evolved so for finding the requisite physical quantities is called the Lebesgue integration.

Thus, the subject "Measure Theory" is the custodian of measurable sets, non-measurable sets, measurable functions and their Lebesgue integration.

Syllabus: Algebras and sigma-algebras, measures, outer measures, measurable sets, Lebesgue measure and its properties, non-measurable sets, measurable functions and their properties, Egoroff's theorem, Lusin's theorem; Lebesgue Integration: simple functions, integral of bounded functions over a set of finite measure, bounded convergence theorem, integral of nonnegative functions, Fatou's lemma, monotone convergence theorem, the general Lebesgue integral, Lebesgue convergence theorem, change of variable formula; Differentiation and integration: functions of bounded variation, differentiation of an integral, absolute continuity; Signed and complex measures, Radon-Nikodym theorem, Lp -spaces and their dual; Product measures, constructions, Fubini's theorem and its applications.

Textbooks/ References:
1. D. L. Cohn, Measure Theory, 1st Edition, Birkhauser, 1994.
2. G. de Barra, Measure Theory and Integration, New Age International, 1981.
3. M. Capinski and E. Kopp, Measure, Integral and Probability, 2nd Edition, Springer, 2007.
4. H. L. Royden, Real Analysis, 3rd Edition, Prentice Hall/Pearson Education, 1988.
5. I. K. Rana, An Introduction to Measure and Integration, Narosa, 1997.