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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.13157 |
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| _version_ | 1866910540795740160 |
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| author | Di Biase, Fausto Krantz, Steven G. |
| author_facet | Di Biase, Fausto Krantz, Steven G. |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13157 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the differentiation of integrals in measure spaces along filters: II Di Biase, Fausto Krantz, Steven G. Functional Analysis Category Theory 28A15, 28A51, 18F99, 20N02 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}. |
| title | On the differentiation of integrals in measure spaces along filters: II |
| topic | Functional Analysis Category Theory 28A15, 28A51, 18F99, 20N02 |
| url | https://arxiv.org/abs/2404.13157 |