The optimal differentiation basis and liftings of 𝐿^{∞}

Abstract

There is an optimal way to differentiate measures when given a consistent choice of where zero limits must occur. The appropriate differentiation basis is formed following the pattern of an earlier construction by the authors of an optimal approach system for producing boundary limits in potential theory. Applications include the existence of Lebesgue points, approximate continuity, and liftings for the space of bounded measurable functions – all aspects of the fact that for every point outside a set of measure 0 0 , a given integrable function has small variation on a set that is “big” near the point. This fact is illuminated here by the replacement of each measurable set with the collection of points where the set is “big”, using a classical base operator. Properties of such operators and of the topologies they generate, e.g., the density and fine topologies, are recalled and extended along the way. Topological considerations are simplified using an extension of base operators from algebras of sets on which they are initially defined to the full power set of the underlying space.