Measurable acting group

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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.

## Definition

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

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

Φ : G × S -> S {\displaystyle \Phi \colon G\times S\to S}

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

## Example: Measurable groups as measurable acting groups

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

## References

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* 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.

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