Measurable acting group

Template:Multiple issues 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.Template:Citation needed

Definition

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

Let $G$ act on $S$ with group action

$\mathrm{\Phi }:G\times S\to S$

If $\mathrm{\Phi }$ is a measurable function from $𝒢\otimes 𝒮$ to $𝒮$, 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 $G$.

References

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