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. Related research deals with covering theorems from geometric measure theory important in many areas of mathematical analysis. I have employed nonstandard analysis methods to optimize the coverings that are used in the Besicovitch Covering Theorem. Together with coauthors, I have simplified the proofs of both the Besicovitch and Morse result. Also with a coauthor I have used the Morse Covering Theorem and the continuity properties of measurable functions to show why the Henstock Integral produces the Lebesgue integral as a limit of Riemann-type sums.