Meas is the category of measurable spaces. The objects are measurable spaces (sets equipped with -algebras), and the morphisms are measurable functions.
Urs: Er, I meant to say here: Meas is the category of measure spaces and measure-preserving maps.
Which terminology should we use? What’s the standard term for the category of measure spaces?
Toby: You can distinguish them as and . But by default, I would expect to mean . (Unless there was some sense that can be built out of and some operator , kind of like how is 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.