Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Looks good.
What can one now do with this suitable category for measure theory? Does it provide interesting universal constructions?
Hi Dmitri, Is it ok if we rename to “enhanced measurable spaces” or “passage to enhanced measurable spaces” or something like that? I am concerned that a newcomer finds this and thinks that they should not use the category of measurable spaces because it has “defects”.
I do understand your points.
But for what I have been doing, it has not always been appropriate to use enhanced measurable spaces. I don’t want to start a fight by writing a page arguing that CSLEMS has defects! but measurable spaces have been more appropriate for some things that I needed to do, and I couldn’t use CSLEMS. In other situations, I have used CSLEMS or variants. I can say more if it helps. Probably other people also have the same issue: it depends what you what to do with it.
I can say more if it helps.
I think it would certainly help to say more here, this is what I meant by feedback. I would be quite interested in seeing more cases that challenge the applicability of CSLEMS.
Edit: I removed the word “defect” altogether and added paragraphs explaining that enhanced measurable spaces are motivated by esthetic considerations and can be easily replaced by measure spaces. I also made it more clear that the article aims to track the existing practices of (conventional) real analysis, probability theory, and statistics, where invariance under equality almost everywhere is required from the beginning.
Re #2: Yes, lots. It is complete and cocomplete, and has a closed monoidal structure. Measurable locales are coreflective in locales.
It will take a while to describe all this structure, though.
Added a disclaimer:
This article concentrates on measure theory as it is used in real analysis, probability theory, statistics, stochastic processes and other areas of analysis. The word “defect” used below is understood as a property that does not match the established practice of these subjects, e.g., identifying functions that are equal almost everywhere. Other areas, such as descriptive set theory may need other criteria and other categories.
Added a remark to clarify that the article is not really about enhanced measurable spaces:
\begin{remark} Everything in this article could be done with conventional measure spaces $(X,M,\mu)$ instead of enhanced measurable spaces. Enhanced measurable spaces are only introduced to enhance the clarity of exposition and avoid making noncanonical choices of measures in some constructions. Also, separating measures from their underlying spaces makes it slightly easier to formulate some theorems, e.g., the Radon–Nikodym theorem. \end{remark}
Eliminated the word “defect” from the article. Rewrote the disclaimer:
This article concentrates on measure theory as it is used in real analysis, probability theory, statistics, stochastic processes and other areas of analysis. In particular, given the existing practice in these fields, we take it for granted that we must identify functions that are equal almost everywhere. Other areas, such as descriptive set theory may need other criteria and other categories.
We also remark that the use of enhanced measurable spaces instead of measure spaces is for strictly esthetic reasons (like avoiding making noncanonical choices of measures), and everything works equally well with traditional measure spaces instead.
Added:
The category $CSLEMS$ has excellent categorical properties: it is complete and cocomplete, admits a closed monoidal structure whose product is the measure-theoretic product, is comonadic over sets and over compact Hausdorff spaces. It also admits a commutative Giry-type probability monad (Furber).
Thanks so much, Dmitri, this looks really nice, it’s a nice survey.
Indeed one area where it might be different is descriptive set theory, I agree. Another, which I’ve been interested in, is in finding nice internal languages, which amount to probabilistic programming languages. There is a very nice language for measurable spaces and the Giry monad, and variations. I’d love to find an internal language for CSLEMS, I think we’ve discussed this before, but I still don’t know how to sort this out, because of wanting to use both tensors. Another possibly related point is that the Markov categories approach does not directly axiomatize CSLEMS, although we can construct things like CSLEMS from a Markov category.
Re #10: Robert Furber’s work constructs a commutative Giry monad on CSLEMS and explains how to get Markov categories from it (see the last two references in the article).
Great, it's nice to have all of the reasons for this category so well summarized!
1 to 13 of 13