Invariant means with special properties
Invariant means with special properties
A finitely additive measure \(m\) on a set \(X\) is said to be continuous if for every \(A \subset X\) there exists \(B \subset X\) such that \(m(A)=2m(B)\). For an amenable group \(G\) acting on \(X\), the author proves that every \(G\)-invariant mean on \(X\) is continuous if and only if the following condition holds: Given any subgroup \(H\) of \(G\) with finite index, there exist \(h_ 1,\dots,h_ n \in H\) satisfying \(F(h_ 1) \cap \cdots \cap F(h_ n)= \emptyset\), where \(F(g)=\{x \in X:gx=x\}\), \(g \in G\).
- University of Perugia Italy
invariant mean, amenable group, Invariant means; Amenability, Means on groups, semigroups, etc.; amenable groups, General groups of measure-preserving transformations, continuous measures
invariant mean, amenable group, Invariant means; Amenability, Means on groups, semigroups, etc.; amenable groups, General groups of measure-preserving transformations, continuous measures
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