On the differentiation of integrals in measure spaces along filters: II

Abstract

Let $X$ be a complete measure space of finite measure. The Lebesgue transform of an integrable function $f$ on $X$ encodes the collection of all the mean-values of $f$ on all measurable subsets of $X$ of positive measure. In the problem of the differentiation of integrals, one seeks to recapture $f$ from its Lebesgue transform. In previous work we showed that, in all known results, $f$ may be recaputed from its Lebesgue transform by means of a limiting process associated to an appropriate family of filters defined on the collection of all measurable subsets of $X$ of positive measure. The first result of the present work is that the existence of such a limiting process is equivalent to the existence of a Von Neumann-Maharam lifting of $X$. In the second result of this work we provide an independent argument that shows that the recourse to filters is a \textit{necessary consequence} of the requirement that the process of recapturing $f$ from its mean-values is associated to a \textit{natural transformation}, in the sense of category theory. This result essentially follows from the Yoneda lemma. As far as we know, this is the first instance of a significant interaction between category theory and the problem of the differentiation of integrals. In the Appendix we have proved, in a precise sense, that \textit{natural transformations fall within the general concept of homomorphism}. As far as we know, this is a novel conclusion: Although it is often said that natural transformations are homomorphisms of functors, this statement appears to be presented as a mere analogy, not in a precise technical sense. In order to achieve this result, we had to bring to the foreground a notion that is implicit in the subject but has remained hidden in the background, i.e., that of \textit{partial magma}.