Timeline for Dense subcategory of measurable spaces
Current License: CC BY-SA 4.0
7 events
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| Jun 5, 2021 at 17:44 | comment | added | Dmitri Pavlov | @MartinBrandenburg: Concerning the category of all measurable spaces, I think if you try the same counterexample (i.e., look at maps to 2^κ, where κ is an arbitrary large infinite cardinal), it works for similar reasons. If κ is greater than the cardinality of all objects in D (or some function thereof), then you are not going to get any nontrivial maps to 2^κ. | |
| Jun 5, 2021 at 17:37 | comment | added | Dmitri Pavlov | @MartinBrandenburg: I meant the paper cited in my answer: arxiv.org/abs/2005.05284. | |
| Jun 5, 2021 at 17:35 | comment | added | Martin Brandenburg | Thanks! Which paper of yours do you mean here? | |
| Jun 5, 2021 at 17:30 | comment | added | Dmitri Pavlov | @MartinBrandenburg: And the category of localizable enhanced measurable spaces (with appropriately defined morphisms) is equivalent to the category of compact strictly localizable enhanced measurable spaces (Remark 5.18 in my paper), although the latter is more convenient by having a simple point-set description of its morphisms. | |
| Jun 5, 2021 at 17:28 | comment | added | Dmitri Pavlov | @MartinBrandenburg: Concerning compactness and strict localizability: all Radon measures satisfy these properties (see Example 4.55 in my paper), as well as many others. Nonlocalizable measurable spaces are extremely pathological: all nontrivial theorems from a typical measure theory textbook fail for them. In particular, the Radon–Nikodym theorem fails, the Riesz representation theorem fails, the duality theorem for L^p spaces fails, etc. So you are not doing measure theory anymore! | |
| Jun 4, 2021 at 20:01 | comment | added | Martin Brandenburg | Thank you, Dmitri! Do you think that there is an answer for the category of all measurable spaces? My question was directed to this category. Can you perhaps also say how restrictive the conditions "compact" and "strictly localizable" are? Are important examples of measurable spaces missing? | |
| Jun 2, 2021 at 23:47 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |