This module covers Lebesgue integration and related topics. It is intended for graduate students in mathematics. Major topics: (1) Quick review of properties of Rn, Lebesgue measure on Rn, Borel sets, Lebesgue nonmeasurable sets, Riemann-Lebesgue function, Lusin’s and Egoroff’s Theorems, convergence in measure. (2) Lebesgue integration, convergence theorems, evaluation of the integral in terms of the distribution function, Lp spaces, density of C�� functions in Lp(Rn), p < ��, abstract integration. (3) Product integration, Fubini’s and Tonelli’s Theorems, application to convolution, approximate identities and maximal function. (4) Lebesgue Differentiation Theorem, Vitali covering, functions of bounded variation, absolutely continuous functions.
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