Fortunately there's a wonderful thing called the "direct integral" which enables one to make sense of direct sums of uncountably infinite families of Hilbert spaces.
Unfortunately I've tried to read about the constructions several times and while every time I walked away with a bit more confidence i'm still not sure how to spell out precisely what properties it satisfies - which is in practice much more important then the construction itself. Hence the question:
Is there a "unique" characterization of the "direct integral" construction for hilbert spaces in terms of the list of the properties it satisfies?
By "unique" I don't mean anything precise, I'm only looking for the most exhaustive list of properties of the construction such that it is more or less clear that the direct integral is not some aribtrary construction.
I'm not asking for a precise categorical definition or anything like that (although if there is one i'd be happy to hear it). I'm just trying to understand the construction enough to be able to use confidently in places where it appears. In particular such a description should be easily adjusted to describe direct integrals of representations of groups/algebras etc...