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 formula_1 a group with group law 
Let further formula_3 be a σ-algebra of subsets of the set formula_4.
The group, or more formally the triple formula_5 is called a measurable group if
Here, formula_12 denotes the formation of the product σ-algebra of the σ-algebras formula_13 and formula_14.
Topological groups as measurable groups.
Every second-countable topological group formula_15 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 formula_17 to formula_17 and from formula_19 to formula_17, respectively. Second countability ensures that formula_21, and therefore the group formula_4 is also a measurable group.
Related concepts.
Measurable groups can be seen as measurable acting groups that act on themselves.