Uniqueness of invariant means for measure-preserving transformations

Abstract:

For some compact abelian groups $X$ (e.g. $T^n$, $n \geqslant 2$, and $\prod \nolimits _{n = 1}^\infty {{Z_2}}$), the group $G$ of topological automorphisms of $X$ has the Haar integral as the unique $G$-invariant mean on ${L_\infty }(X,{\lambda _X})$. This gives a new characterization of Lebesgue measure on the bounded Lebesgue measurable subsets $\beta$ of ${R^n}$, $n \geqslant 3$; it is the unique normalized positive finitely-additive measure on $\beta$ which is invariant under isometries and the transformation of ${R^n}:({x_1}, \ldots ,{x_n}) \mapsto ({x_1} + {x_2},{x_2}, \ldots ,{x_n})$. Other examples of, as well as necessary and sufficient conditions for, the uniqueness of a mean on ${L_\infty }(X,\beta ,p)$, which is invariant by some group of measure-preserving transformations of the probability space $(X,\beta ,p)$, are described.