Measurable group

Let ${\displaystyle (G,\circ )}$ a group with group law

${\displaystyle \circ :G\times G\to G}$.

Let further ${\displaystyle {\mathcal {G}}}$ be a σ-algebra of subsets of the set ${\displaystyle G}$.

The group, or more formally the triple ${\displaystyle (G,\circ ,{\mathcal {G}})}$ is called a measurable group if[1]

• the inversion ${\displaystyle g\mapsto g^{-1}}$ is measurable from ${\displaystyle {\mathcal {G}}}$ to ${\displaystyle {\mathcal {G}}}$.
• the group law ${\displaystyle (g_{1},g_{2})\mapsto g_{1}\circ g_{2}}$ is measurable from ${\displaystyle {\mathcal {G}}\otimes {\mathcal {G}}}$ to ${\displaystyle {\mathcal {G}}}$

Here, ${\displaystyle {\mathcal {A}}\otimes {\mathcal {B}}}$ denotes the formation of the product σ-algebra of the σ-algebras ${\displaystyle {\mathcal {A}}}$ and ${\displaystyle {\mathcal {B}}}$.

Every second-countable topological group ${\displaystyle (G,{\mathcal {O}})}$ can be taken as a measurable group. This is done by equipping the group with the Borel σ-algebra

${\displaystyle {\mathcal {B}}(G)=\sigma ({\mathcal {O}})}$,

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 ${\displaystyle {\mathcal {B}}(G)}$ to ${\displaystyle {\mathcal {B}}(G)}$ and from ${\displaystyle {\mathcal {B}}(G\times G)}$ to ${\displaystyle {\mathcal {B}}(G)}$, respectively. Second countability ensures that ${\displaystyle {\mathcal {B}}(G)\otimes {\mathcal {B}}(G)={\mathcal {B}}(G\times G)}$, and therefore the group ${\displaystyle G}$ is also a measurable group.

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