Ergodic group actions (Q1070051)

Let G be a countable group. Suppose \(F\subset G\) generates G. Let u be a probability measure whose support is F. Let \((v_ n)\) be any sequence of finitely-supported probability measures on G such that \(\| v_ n- (1/n)\sum^{n}_{k=1}u^ k\|_ 1<1/2^ n.\) Then for any ergodic action of G as measure-preserving transformations of a probability space (X,\(\beta\),m), if \(F\in L_ p(X)\), \(1\leq p<\infty\), the averages \((v_ nF)\) converge a.e. and in norm to \(\int F dm\). This shows how to explicitly construct summing sequences for general group actions if one allows weighted averages. Further details of this construction, improvements in the case that G has Khazdan's property T, and the possibility of extension of this result to uniquely ergodic systems are also discussed.