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

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Let   be a measurable group, where   denotes the  -algebra on   and   the group law. Let further   be a measurable space and let   be the product  -algebra of the  -algebras   and  .

Let   act on   with group action

 

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

Example: Measurable groups as measurable acting groups

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

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.