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.
be a σ-algebra of subsets of the set
The group, or more formally the triple
is called a measurable group if[1] Here,
denotes the formation of the product σ-algebra of the σ-algebras
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
Second countability ensures that
Measurable groups can be seen as measurable acting groups that act on themselves.