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Is there a common notation in the literature for

  • the category of measurable spaces and measurable maps?
  • the category of measure spaces and measure-preserving maps?

The nlab suggests $\mathsf{Measble}$ for the category of measurable spaces, but this looks a little bit ugly, and the nlab doesn't suggest something for the category of measure spaces. If there is no common notation, what is your suggestion? For example, $\mathsf{Meas}$ looks fine, but it is not clear a priori which category this should denote.

Added. According to the comments, $\mathsf{Meas}$ is a common notation for the category of measurable spaces and measurable maps. So what about the category of measure spaces? If there is no common notation: What do you think about $\mathsf{MeasSp}$, $\mathsf{MeaSp}$, or $\mathsf{Measure}$? An argument for the latter: Rudin points out in his book on real and complex analysis that the whole information of a measure space $(\Omega,\mathcal{A},\mu)$ is already encoded in the measure $\mu$, since $\mathcal{A}$ is the domain of $\mu$ and $\Omega$ is the greatest element of $\mathcal{A}$.

Background. I'm writing a text on category theory and thereby have found a nice example of a natural transformation: Consider the category $\mathcal{C}$ of measure spaces and measure-preserving maps and the category $\mathsf{Ban}$ of Banach spaces with non-expansive linear maps. Then $L^1 : \mathcal{C}^{\mathrm{op}} \to \mathsf{Ban}$ is a functor and the integral provides us with a natural transformation $\int: L^1 \to \Delta(\mathbb{R})$. The naturality is precisely the general transformation formula. Wouldn't it be nice to give $\mathcal{C}$ a unique name? By the way, a similar natural transformation is used in Hartig's paper "The Riesz Representation Theorem Revisited" to give a conceptual proof of the Riesz Representation Theorem.

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    $\begingroup$ I have used the nLab's notation with disappointment for lack of a better alternative. $\endgroup$ Mar 24, 2015 at 15:44
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    $\begingroup$ I'm familiar with meas for the first category. See for example here. $\endgroup$ Mar 24, 2015 at 17:36
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    $\begingroup$ How does this natural transf. relate to Tom Leinster's characterisation of Lebesgue integral? $\endgroup$
    – David Roberts
    Mar 25, 2015 at 10:17
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    $\begingroup$ @MartinBrandenburg: The former. When trying to prove any nontrivial theorem in measure theory (e.g., Riesz, Radon-Nikodym, etc.), one immediately runs into necessity of using null sets, so measurable spaces do not suffice. On the other hand, measured spaces have too much data, which ruins pretty much every categorical construction (e.g., finite products). But one can define an intermediate category (σ-algebras with σ-ideals of null sets) that has very good categorical properties and easily fits existing classical theorems of measure theory such as the ones mentioned above. $\endgroup$ Mar 25, 2015 at 10:43
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    $\begingroup$ @MartinBrandenburg: This is described in more detail in mathoverflow.net/a/20820 and ncatlab.org/nlab/show/measurable+locale. $\endgroup$ Mar 25, 2015 at 10:45

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You can consider the notation Bor or Borel for the category of measurable (often called Borel) spaces.

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  • $\begingroup$ Thank you! I didn't know this alternative terminology. "measurable space" seems to be more modern than "Borel space". Is this correct? $\endgroup$ Mar 25, 2015 at 22:21
  • $\begingroup$ Actually $\mathsf{Bor}$ looks really good. But then it's still open how to denote the category of measure spaces. (Or the category of measure spaces equipped with a $\sigma$-ideal, as suggested by Dmitri Pavlov.) $\endgroup$ Mar 26, 2015 at 7:34
  • $\begingroup$ It's measurable spaces that need to be equipped with a $\sigma$-ideal pace Pavlov (measure spaces already are). $\endgroup$ Apr 5, 2017 at 19:52

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