In mathematics, a measurable group is a special type of group in the intersection between group theory and measure theory. Measurable groups are used to study measures is an abstract setting and are often closely related to topological groups.

Definition

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Let   a group with group law

 .

Let further   be a σ-algebra of subsets of the set  .

The group, or more formally the triple   is called a measurable group if[1]

  • the inversion   is measurable from   to  .
  • the group law   is measurable from   to  

Here,   denotes the formation of the product σ-algebra of the σ-algebras   and  .

Topological groups as measurable groups

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Every second-countable topological group   can be taken as a measurable group. This is done by equipping the group with the Borel σ-algebra

 ,

which is the σ-algebra generated by the topology. Since by definition of a topological group, the group law and the formation of the inverse element is continuous, both operations are in this case also measurable from   to   and from   to  , respectively. Second countability ensures that  , and therefore the group   is also a measurable group.

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Measurable groups can be seen as measurable acting groups that act on themselves.

References

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  1. ^ Kallenberg, Olav (2017). Random Measures, Theory and Applications. Switzerland: Springer. p. 266. doi:10.1007/978-3-319-41598-7. ISBN 978-3-319-41596-3.