Complex Analysis the Hard Way: a Measure Theoretical Approach

Abstract

In this thesis we present a sophisticated, non-standard treatment of complex analysis using modern tools from measure theory and advanced analysis. This approach opens the path to deep and powerful results, including the regularity theorem and the global solution to the inhomogeneous Cauchy-Riemann equation on a disc and on the complex plane. Moreover, the resulting theory is suitable for generalisation to the further study of Riemann surfaces, and of complex geometry more generally.