Measurable group

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

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

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

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.

Related concepts

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