Uniformly cyclic vectors
A group acting on a measure space $(X,\beta,\lambda)$ may or may not admit a cyclic vector in $L_\infty(X)$. This can occur when the acting group is as big as the group of all measure-preserving transformations. But it does not occur, even though there is no cardinality obstruction to it, for the regular action of a group on itself. The connection of cyclic vectors to the uniqueness of invariant means is also discussed.