# 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 $f:\Omega \to \Delta$ , where $(\Omega ,{\mathcal {F}},\mu )$ and $(\Delta ,{\mathcal {G}},\lambda )$ are measure spaces, is called measurable if and only if for each $D\in {\mathcal {G}}$ , we have $f^{-1}(D)\in {\mathcal {F}}$ .

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 $(\Omega ,{\mathcal {F}},\mu )$ and $(\Delta ,{\mathcal {G}},\lambda )$ be measure spaces. An algebra map from $(\Omega ,{\mathcal {F}},\mu )$ to $(\Delta ,{\mathcal {G}},\lambda )$ is a function $\phi :{\mathcal {F}}\to {\mathcal {G}}$ such that $\lambda (\phi (A))=\mu (A)$ for all $A\in {\mathcal {F}}$ , and moreover, for $A,A_{1},A_{2},\ldots \in {\mathcal {F}}$ 1. $\phi (\emptyset )=\emptyset$ and $\phi (\Omega )=\Delta$ 2. $\phi \left(\bigcup _{n\in \mathbb {N} }A_{n}\right)=\bigcup _{n\in \mathbb {N} }\phi (A_{n})$ 3. $\phi (\Omega \setminus A)=\Delta \setminus \phi (A)$ .

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 $f:\Omega \to \Delta$ , where $(\Omega ,{\mathcal {F}},\mu )$ and $(\Delta ,{\mathcal {G}},\lambda )$ are measure spaces, is called measure-preserving if and only if it is measurable and $\mu (f^{-1}(D))=\lambda (D)$ for all $D\in {\mathcal {G}}$ .

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.