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.