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Journal Article
On the Differentiation of Integrals in Measure Spaces Along Filters: II
2024
Overview
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
.
Publisher
Springer International Publishing,Springer Nature B.V
Subject
