Graduate Analysis I

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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