Measurable acting group

In mathematics, a measurable acting group is a special group that acts on some space in a way that is compatible with structures of measure theory. Measurable acting groups are found in the intersection of measure theory and group theory, two sub-disciplines of mathematics. Measurable acting groups are the basis for the study of invariant measures in abstract settings, most famously the Haar measure, and the study of stationary random measures.

Definition
Let $$ (G, \mathcal G, \circ) $$ be a measurable group, where $$ \mathcal G $$ denotes the $ \sigma $-algebra on $$ G $$ and $$ \circ $$ the group law. Let further $$ (S, \mathcal S) $$ be a measurable space and let $$ \mathcal A \otimes \mathcal B $$ be the product $ \sigma $-algebra of the $$ \sigma $$-algebras $$ \mathcal A $$ and $$ \mathcal B $$.

Let $$ G $$ act on $$ S $$ with group action
 * $$ \Phi \colon G \times S \to S $$

If $$ \Phi $$ is a measurable function from $$ \mathcal G \otimes \mathcal S $$ to $$ \mathcal S $$, then it is called a measurable group action. In this case, the group $$ G $$ is said to act measurably on $$ S $$.

Example: Measurable groups as measurable acting groups
One special case of measurable acting groups are measurable groups themselves. If $$ S=G $$, and the group action is the group law, then a measurable group is a group $$ G $$, acting measurably on $$ G $$.