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

Let [math]\displaystyle{ G }[/math] act on [math]\displaystyle{ S }[/math] with group action

[math]\displaystyle{ \Phi \colon G \times S \to S }[/math]

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

## Example: Measurable groups as measurable acting groups

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

## 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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