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

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 ${\displaystyle (\Omega ,{\mathcal {F}},\mu )}$ and ${\displaystyle (\Delta ,{\mathcal {G}},\lambda )}$ be measure spaces. An algebra map from ${\displaystyle (\Omega ,{\mathcal {F}},\mu )}$ to ${\displaystyle (\Delta ,{\mathcal {G}},\lambda )}$ is a function ${\displaystyle \phi :{\mathcal {F}}\to {\mathcal {G}}}$ such that ${\displaystyle \lambda (\phi (A))=\mu (A)}$ for all ${\displaystyle A\in {\mathcal {F}}}$, and moreover, for ${\displaystyle A,A_{1},A_{2},\ldots \in {\mathcal {F}}}$

1. ${\displaystyle \phi (\emptyset )=\emptyset }$ and ${\displaystyle \phi (\Omega )=\Delta }$
2. ${\displaystyle \phi \left(\bigcup _{n\in \mathbb {N} }A_{n}\right)=\bigcup _{n\in \mathbb {N} }\phi (A_{n})}$
3. ${\displaystyle \phi (\Omega \setminus A)=\Delta \setminus \phi (A)}$.

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

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.