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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 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.