Measurable acting group

In mathematics, a measurable acting group is a special group that acts on some space in a way that is compatible with structures of measure theory. Measurable acting groups are found in the intersection of measure theory and group theory, two sub-disciplines of mathematics. Measurable acting groups are the basis for the study of invariant measures in abstract settings, most famously the Haar measure, and the study of stationary random measures.^{[citation needed]}

Definition

Let ${\displaystyle (G,{\mathcal {G}},\circ )}$ be a measurable group, where ${\displaystyle {\mathcal {G}}}$ denotes the ${\displaystyle \sigma }$-algebra on ${\displaystyle G}$ and ${\displaystyle \circ }$ the group law. Let further ${\displaystyle (S,{\mathcal {S}})}$ be a measurable space and let ${\displaystyle {\mathcal {A}}\otimes {\mathcal {B}}}$ be the product ${\displaystyle \sigma }$-algebra of the ${\displaystyle \sigma }$-algebras ${\displaystyle {\mathcal {A}}}$ and ${\displaystyle {\mathcal {B}}}$.

Let ${\displaystyle G}$ act on ${\displaystyle S}$ with group action

${\displaystyle \Phi \colon G\times S\to S}$

If ${\displaystyle \Phi }$ is a measurable function from ${\displaystyle {\mathcal {G}}\otimes {\mathcal {S}}}$ to ${\displaystyle {\mathcal {S}}}$, then it is called a measurable group action. In this case, the group ${\displaystyle G}$ is said to act measurably on ${\displaystyle S}$.

Example: Measurable groups as measurable acting groups

One special case of measurable acting groups are measurable groups themselves. If ${\displaystyle S=G}$, and the group action is the group law, then a measurable group is a group ${\displaystyle G}$, acting measurably on ${\displaystyle G}$.

References

• Kallenberg, Olav (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling. Vol. 77. Switzerland: Springer. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.