Consequently, researchers including myself have obtained new results with these methods in areas such as potential theory, stochastic processes, the study of equilibria in mathematical economics, and the study in mathematical physics of infinite particle systems and gas kinetics. My research also treats problems of representing measures and ideal boundaries in potential theory. A recently developed technique in this work reduces to a special case the production of Radon-Nikodym derivatives from various limiting processes. This result casts new light on the martingale convergence theorem and significantly simplifies the treatment of fine limit theorems in potential theory and measure differentiation theorems. Indeed, previous results in the literature suggested that no such system could exist.
Analogous results produce an optimal differentiation basis for measure differentiation.
Associated with this work is a simplified treatment of b ase operators. These are simple, fundamental operators in analysis, mapping sets to sets. A base operator generates a topology for which a set C is closed if C contains its image under the operator. Examples are the fine topology in potential theory and the density topology in measure theory. Part of this work is an investigation of the microscopic behavior of measurable functions and an appropriate selection of representatives from the a.e.-equivalence classes they form.