Amenable discrete group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

A group $G$ is termed an amenable discrete group if there exists a function that assigns to each subset of $G$ a real number in $[0,1]$ such that:

• The number associated with the whole of $G$ is $1$
• The function is finitely additive on subsets. In other words, the number associated with a disjoint union of countably many subsets is the sum of the numbers associated with each of the subsets
• The function is left-invariant, viz the value associated with a subset $S$ is the same as the value associated with $gS$

In other words, we have a measure of the Borel $\sigma$-algebra of all subsets of the given discrete group.

Often, when dealing with groups abstractly, we use the term amenable group for amenable discrete group (in other words, we assume that the discrete topology is put over the group).

Relation with other properties

Stronger properties

• Finite group, equipped with the counting measure. This is the measure associating to every subset the ratio of its cardinality to the cardinality of the group
• Solvable group: For full proof, refer: Solvable implies amenable

Metaproperties

Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other finite direct product-closed group properties

A finite direct product of amenable discrete groups is again an amenable discrete group.