4

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.

Martin Brandenburg
  • 61,443
  • 2
    I have used the nLab's notation with disappointment for lack of a better alternative. – Steve Huntsman Mar 24 '15 at 15:44
  • 1
    I'm familiar with meas for the first category. See for example here. – Michael Greinecker Mar 24 '15 at 17:36
  • @MichaelGreinecker: Thanks! The authors even say "This gives a category which is often called $\mathsf{Meas}$.". So it should be common. Have you seen a notation for the other category? – Martin Brandenburg Mar 24 '15 at 19:03
  • @MartinBrandenburg Sorry, I haven't. – Michael Greinecker Mar 24 '15 at 20:56
  • What do you think about $\mathsf{Measure}$ for the category of measure spaces (and $\mathsf{Meas}$ for the category of measurable spaces)? – Martin Brandenburg Mar 25 '15 at 08:58
  • I don't think there is any commonly used notation for these categories. One reason for that might be that it's hard to do any nontrivial measure theory with these categories. – Dmitri Pavlov Mar 25 '15 at 09:49
  • 1
    @DmitriPavlov: Thank you for your comment. I have found several papers on measure theory and probability theory which apply category theory to the categories of measurable resp. probability spaces. Do you say that it is hard to do nontrivial measure theory with these categories, but that it is possible for other categories, or that category theory is not useful for nontrivial measure theory at all? In the former case, I would like to know which alternative categories are more suitable. – Martin Brandenburg Mar 25 '15 at 09:59
  • 2
    How does this natural transf. relate to Tom Leinster's characterisation of Lebesgue integral? – David Roberts Mar 25 '15 at 10:17
  • 5
    @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. – Dmitri Pavlov Mar 25 '15 at 10:43
  • 3
    @MartinBrandenburg: This is described in more detail in http://mathoverflow.net/a/20820 and http://ncatlab.org/nlab/show/measurable+locale. – Dmitri Pavlov Mar 25 '15 at 10:45
  • @MartinBrandenburg: I think Wikipedia's choice to call this the change-of-variables-formula is somewhat misguided. It's a useful formula, I give you that, but "change of variables" usually refers to the theorem that actually computes the push-forward measure in terms of the Jacobian. – Johannes Hahn Mar 25 '15 at 14:46
  • @JohannesHahn: Yes, I would also call it the (general) 'transformation formula'. – Martin Brandenburg Mar 25 '15 at 22:19
  • 1
    Another useful category related to this discussion is the one where objects are measurable spaces (in whatever meaning) and morphisms are Markov kernels, i.e. measurable mappings from the first object to probability measures on the second object. There is a natural definition of a measure preserving kernel, so there is also another useful category where the objects are probability spaces and morphisms are Markov kernels. –  Mar 25 '15 at 14:28
  • I seem to have introduced ‘Measble’ to the nLab, and I probably did it because I always thought that ‘Meas’ was for measure spaces instead. – Toby Bartels Apr 05 '17 at 20:06

1 Answers1

4

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

Todd Trimble
  • 52,336
Brius
  • 41
  • Thank you! I didn't know this alternative terminology. "measurable space" seems to be more modern than "Borel space". Is this correct? – Martin Brandenburg Mar 25 '15 at 22:21
  • 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.) – Martin Brandenburg Mar 26 '15 at 07:34
  • It's measurable spaces that need to be equipped with a $\sigma$-ideal pace Pavlov (measure spaces already are). – Toby Bartels Apr 05 '17 at 19:52