Measure Theory/Morphisms and categories of measure spaces

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