nLab
Meas

Meas is the category of measurable spaces. The objects are measurable spaces (sets equipped with $\sigma$-algebras), and the morphisms are measurable functions.

Discussion

Urs: Er, I meant to say here: Meas is the category of measure spaces and measure-preserving maps.

But now I recall that John uses Meas for a 2-categorical version of that.

Which terminology should we use? What’s the standard term for the category of measure spaces?

Toby: You can distinguish them as $MeasSp$ and $MeasCat$. But by default, I would expect $Meas$ to mean $MeasSp$. (Unless there was some sense that $MeasSp$ can be built out of $Sp$ and some operator $Meas$, kind of like how $GrayCat$ is $Gray‑Cat$ for suitable Gray. But life is probably not that beautiful.)

Tom: The category of measured spaces is notoriously difficult to work with from the structural point of view. I suggest that Meas refer to the more basic category whose objects are measurable spaces and morphisms are measurable functions.

category: category