There are several categories whose objects are measure spaces. Naturally, they are determined by a choice of morphisms.
Definition (measurable):
A function , where and are measure spaces, is called measurable if and only if for each , we have .
Definition (standard category of measure spaces):
The standard category of measure spaces is the category whose objects are measure spaces and whose morphisms are the measurable functions.
Definition (algebra map):
Let and be measure spaces. An algebra map from to is a function such that for all , and moreover, for
- and
- .
Definition (algebra map category of measure spaces):
The algebra map category of measure spaces is the category whose objects are measure spaces and whose morphisms are algebra maps.
Definition (measure-preserving):
A function , where and are measure spaces, is called measure-preserving if and only if it is measurable and for all .
Definition (alternative category of measure spaces):
The standard category of measure spaces is the category whose objects are measure spaces and whose morphisms are the measure-preserving functions.