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# Real Analysis I

Syllabus

Lebesgue outer measure, introduction and basic properties, Lebesgue measurable sets, basic properties, Lebesgue measure, properties of Lebesgue measure, further characterizations of Lebesgue measurable sets, Existence of a non-measurable set using Axiom of Choice, a brief introduction to the notion of inner measure, some basic properties. Lebesgue measurable functions, definition and basic properties, simple functions, measurable functions as limit of sequences of simple functions, convergence of sequences of measurable functions, convergence almost everywhere, convergence in measure and almost uniform convergence, basic properties, interdependence, Egoroffs Theorem, Lusins Theorem. Lebesgue integration, introduction to the notion of Lebesgue integral of bounded measurable functions, basic properties, Bounded convergence Theorem, relation between Riemann integration and Lebesgue integration of bounded functions. Lebesgue integration of non-negative but arbitrary measurable functions, basic properties, Monotone convergence Theorem, Fataus Lemma. Lebesgue integration of arbitrary measurable functions, Dominated convergence Theorem. Functions of bounded variations, basic properties, absolutely continuous functions, basic properties, invariance of measurability under absolutely continuous functions, Vitali covering Theorem, differentiation of monotone non-decreasing functions, functions of bounded variations and absolutely continuous functions. Revisiting Fundamental Theorem of Integral Calculus, Version for the Riemann integration, Example of its deficiency, Fundamental theorem of Integral Calculus for Lebesgue integration, existence of anti-derivatives, some further results. Abstract measure theory, ring, sigma ring, algebra, sigma algebra, monotone class, basic properties and set theoretic results. Introduction of the notion of measure on a ring, basic properties, notion of outer measure on a hereditary sigma ring, measurable sets, extension of measure, unicity of the extension. Measurable cover, complete measure and completion of a measure. Riemann Stieltjes integral, basic properties, Darbaux-Stieltjes integral, relation between the two concepts, Riemann condition, further results involving increasing integrators, Mean Value Theorems, l_p and L_p Spaces and Integral Operators.

Analysis I & II

Point Set Topology

Projects 15%

Midterm 35%

Final 50%

N/A

Time: Saturday and Monday from 13:00 to 15:00

**Place: **Calss 9 in College of Mathematics Building