On the Differentiation of Integrals in Measure Spaces Along Filters

It is known that the study of the boundary behavior of (harmonic or) holomorphic functions, to which N. Sibony has contributed with penetrating work, is linked to the differentiation of integrals. In 1936, R. de Possel observed that, in the general setting of a measure space with no metric structure, certain phenomena, relative to the differentiation of integrals, which are familiar in the Euclidean setting precisely because of the presence of a metric, are devoid of actual meaning. In the first part of this work, we introduce the concept of functional convergence class that provides a unifying framework for various limiting processes and enables us to establish a hierarchy between them, and show that, within this hierarchy, the notion of filter (introduced by H. Cartan just a year after De Possel’s contribution) occupies the position of wider scope. In the second part of this work, we show how to reformulate some of the contributions of de Possel in the language of filters.