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.

## Definition

Let (G,𝒢,∘) be a measurable group, where 𝒢 denotes the σ-algebra on G and ∘ the group law. Let further (S,𝒮) be a measurable space and let 𝒜⊗ℬ be the product σ-algebra of the σ-algebras 𝒜 and ℬ.

Let G act on S with group action

Φ:G×S->S

If Φ is a measurable function from 𝒢⊗𝒮 to 𝒮, then it is called a measurable group action. In this case, the group G is said to act measurably on S.

## Example: Measurable groups as measurable acting groups

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

## References

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

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