On the Differentiation of Integrals in Measure Spaces Along Filters: II

• Published in: Complex analysis and operator theory Vol. 18; no. 5
• Main Authors: Di Biase, Fausto, Krantz, Steven G.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.07.2024; Springer Nature B.V
• Subjects: Analysis; Constraining; Differentiation; Homomorphisms; Integrals; Magma; Mathematical functions; Mathematics; Mathematics and Statistics; Operator Theory; Filters; 18F99; Differentiation of integrals; Natural transformations; 28A15; Almost everywhere convergence; Liftings; 20N02; Partial magmas; Moore-Smith sequences; The Yoneda lemma; 28A51
• ISSN: 1661-8254; 1661-8262
• DOI: 10.1007/s11785-024-01552-y

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 recaptured from its Lebesgue transform by means of a limiting process associated to an appropriate family of filters defined on the collection A 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 necessary consequence of the requirement that the process of recapturing f from its mean-values is associated to a 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 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 partial magma .