This question is naive, but I didn't get an answer at [MSE](https://math.stackexchange.com/questions/4587217/is-there-a-measure-theory-for-proper-classes): Is it straightforward to extend measure theory to proper classes?


Of course when one tries to define measures on "large sets" problems of non-measurability arise (e.g. non-Lebesgue measurable sets, Ulam's theorem, etc.). But setting those aside (since they arise already for measures over sets), are there any special problems that dealing with proper classes would raise?


To be a little bit more precise (and I'm afraid I can't do any better at the moment), let $\Omega$ be a proper class. If you prefer that we have something concrete, then let $\Omega$ be the proper class of all ordinals, for instance. Is there anything wrong with saying that a measure is a real-valued function with the usual properties (countable additivity, etc.) on a collection of sub-classes of $\Omega$?


In other words, can the usual definition of a measure be extended straightforwardly to proper classes or are there problems lurking here?


If this isn't entirely straightforward and you know of any references, I would appreciate them.



Is there a measure theory for proper classes?
As a [sledgehammer](https://www.collinsdictionary.com/us/dictionary/english/a-sledgehammer-to-crack-a-nut), working in Higher Order Set Theory (HOST) as proposed in


* *Rhea, Alec*, An Axiomatic Approach to Higher Order Set Theory. (2022)
<https://doi.org/10.48550/arXiv.2206.10060>


you can just go ahead and collect up proper classes inside other proper classes, no need to move higher up the hierarchy or inevitable collapse downwards due to membership issues. I designed this theory to work with things like topology, analysis, measure theory, etc. over the surreals without having to worry about anything too subtle, and it works equally well for measure theory on arbitrary proper classes -- no subtleties lurking about.


This theory is equiconsistent with ZFC plus a countable collection of inaccessible cardinals, however, so it is significantly stronger in consistency strength than most standard set theories. If there's anything you'd like me to elaborate on please don't hesitate to ask.


