Advanced Topics in Real Analysis

Complete a modern and robust foundation of measure theory and integration and differentiability of functions begun in Mathematical Analysis 4 and Real Analysis. In particular, the extension of the fundamental theorems of integral calculus to weakly differentiable vector fields and rough sets. In addition the study of differentiability of convex functions and their approximation by semicontinuous functions, which is the basis of viscosity methods for fully nonlinear partial differential equations.

Capacity to apply the theorems of Radon-Nikodym and weak compactness for Radon Measures. Capacity to verify the validity and to apply integration by parts formulas for weakly differentiable functions on rough sets. Capacity to reduce questions of differentiability to exceptional sets of zero measure.