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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.[ citation needed ]
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 .
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 .