| The aim of the course |
The aim of this course is to give the principle theory of real valued functions, study the concepts of metric spaces,complete ,continuty , compactness, connectedness,inner measure and outer measure of a set, infinite sets,measurable sets, measurable functions, Lebesque integral,space of integrable funtions and their applications.
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| Course Content |
This course covers metric spaces,complete, continuity, and compactness, connectedness, principle of inverse transformation, infinite sets ,countable infinity ,continuum , point sets,Limit point, closed sets, open sets and structure of theses sets,accumulation points,Borel sets, measurable sets, the measure of open sets, the measure of closed sets inner and outer measure of bounded sets, the class of measurable sets, Vitali's theorem and its consequences , measurable functions, their properties, Lebesque measure, Lebesgue measure space ,null measure sets , step functions, Lebesgue's intergral on measurable set , reconstruction of primitive function, Lebesque integral of bounded functions and its properties , comparing Lebesgue integral with Riemann integral , convergence and theorems of convergence , typs of convergence , Fubini's theorem and its consequences , square integrable functions, Lp spaces, and its properties, Riemann-Stieltijes integral and its properties , Lebesgue Stieltijes integral and properties.
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