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 formula_1 be a measurable group, where formula_2 denotes the formula_3-algebra on formula_4 and formula_5 the group law. Let further formula_6 be a measurable space and let formula_7 be the product formula_3-algebra of the formula_3-algebras formula_10 and formula_11.
Let formula_4 act on formula_13 with group action
If formula_15 is a measurable function from formula_16 to formula_17, then it is called a measurable group action. In this case, the group formula_4 is said to act measurably on formula_13.
Example: Measurable groups as measurable acting groups.
One special case of measurable acting groups are measurable groups themselves. If formula_20, and the group action is the group law, then a measurable group is a group formula_4, acting measurably on formula_4.