Advanced Topics in Real Analysis

-- Measure theory: review of measure theory. Borel and Radon measures in Euclidean spaces. Measurable functions and the the theorems of Lusin and Egoroff. Review integrazion of masurable functions emphasizing Lebesgue and Hausdorff measures. The covering theorems of Vitali and Besicovitch. Differentiation of Radon measures, the Radon-Nikodym theorem, and Lebesgue's differentiation theorem. The Riesz representation theorem for Radon measures, weak convergence and weak compactness for Radon measures.

-- Area and coarea fromulas: Lipschitz functions and Rademacher's theorem. Linear maps and the Jacobian of Lischitz maps. Area and coarea formulas.

-- Functions of bounded variation (BV) in several variables and sets of finite perimeter: Definitions of BV functions and sets of finite perimeter, structure theorem for BV functions. Approximation and compactness for BV functions. Boundary traces and extensions for BV functions. Coarea formula, isoperimetric inequality and relations between capacity and Hausdorff measure. The reduced boundary and structure theorem for sets of finite perimeter. The divergence theorem for BV vector fields on sets of finite perimeter.

-- Differentiation of convex functions in several variables: the subdifferential and the first order theory. Weak derivatives of first and second order. Alexandroff's theorem on the second order differentiability in the sense of Peano. Sub and superdifferentials of second order, semicontinuous functions and semiconvex functions.