# 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.

**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.