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 ![{\displaystyle \phi (\Omega )=\Delta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c355bed813944fa8d520d302b23de51c8967b239)
![{\displaystyle \phi \left(\bigcup _{n\in \mathbb {N} }A_{n}\right)=\bigcup _{n\in \mathbb {N} }\phi (A_{n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e632be7c16df2e3815115c3d0bbf2c55381856ba)
.
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.