Change of base for measure spaces
Interesting applications of category theory to several topics of analysis have been attained by certain authors, but there has been little study of Hilbert spaces and their measure theory. Motivated by the measure-indexed nature and the coproduct-like aspect of the direct integral of Hilbert spaces, the author's aim is to interpret this construction in an indexed categorical setting, now. To this end, two monoidal categories of finite measure spaces relevant to indexing introduced elsewhere are still considered: \({\mathcal M}{\mathcal O}{\mathcal R}\) with morphisms the zero-reflecting measurable functions; and \({\mathcal D}isint\) with morphisms called disintegrations, i.e., measurable functions satisfying two special conditions. Both have various properties but products are lacking. Anyway, as disintegrations have a ``built-in self-indexing'' nature, the premise is that an object of \({\mathcal D}isint/X\) represents a notion proper to an \(X\)-family of measure spaces. For a measurable family of Hilbert spaces over a chosen object \(X\) in \({\mathcal D}isint\), however, three additional axioms on the objects of the slice category \({\mathcal M}ble/X\) are required, the so resulting subcategory is denoted by \({\mathcal H}{\mathcal F}/X\). The associated ``direct integral'' functor \(\int^\oplus:{\mathcal H}{\mathcal F}/X\to{\mathcal H}ilb\) is therefore well defined, in the sense that the image of an object under it is actually complete. In the epilogue some questions and scholia on the subject are formulated.
scientific article; zbMATH DE number 1308176