Subsequence ergodic theorems for amenable groups (Q1802755)

Let \(G\) be an amenable group and let \(\{A_ n\}\) be a Følner sequence of subsets \(A_ n\) of \(G\) such that \[ \lim \sup {| A_ n^{-1} A_ n | \over | A_ n |} < \infty. \] Let \((X, {\mathcal B}, \mu,G)\) be an ergodic action and \(b\) a bounded function of \(X\). Under these assumptions the authors prove that almost every \(x \in X\) has the following property: for any measure preserving \((Y, {\mathcal C}, \nu,G)\) and \(f \in L^ 1(Y, {\mathcal C}, \nu)\) the sequence \[ {1 \over | A_ n |} \sum_{g \in A_ n} b(gx) f(gy) \] converges for \(\nu\)-a.e. \(y \in Y\).