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The category of disintegration. (English) Zbl 0818.18003
Regarding finite measure spaces, measure preserving maps are very stringent, in fact these are too much like equivalences, while the measurable maps constitute a category apparently similar, but different to $${\mathcal T}op$$. Here two monoidal categories are presented to compensate for the lack of products therein; the morphisms are chosen to be the measure zero reflecting (measurable) maps in the former, and the so- called disintegrations in the latter. They are proposed as a background to an ultimate study, in the sequel, of abstract $$X$$-families in the context of indexed category theory. The Radon-Nikodym derivate and Fubini’s theorem are also encoded in a categorical vein, in specific examples [compare with F. E. J. Linton, Functorial measure theory, Funct. Anal., Proc. Conf., Univ. Calif., Irvine 1966, 36-49 (1967; Zbl 0218.28006)].

##### MSC:
 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 28E05 Nonstandard measure theory 46G05 Derivatives of functions in infinite-dimensional spaces
Zbl 0218.28006
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##### References:
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