Ergodic theorems for convolutions of a measure on a group
Let \(T\) be a representation of a locally compact Hausdorff group \(G\) as measure-preserving transformations of a \(\sigma\)-finite measure space \((X, m)\). Then any probability measure \(\mu\) on \(G\) induces an operator \(T_\mu\) on \(L_p(X, m)\), \(p\geq 1\) (such that \(\langle T_\mu f_1, f_2\rangle= \int \langle T_g f_1, f_2\rangle d\mu(g)\) for all \(f_1\in L_p(X, m)\) and \(f_2\in L_q(X, m)\), \(q\) being the conjugate of \(p\)). The paper under review is concerned with the question of convergence of \(\{T^n_\mu f\}\), \(f\in L_p\), in \(L_p\)-norm and almost everywhere, which has been studied by many authors under various conditions. Several new results are proved on the question, some of which generalize previous results to a more general setting. We will not go into the details of the results, which are varied and somewhat technical. It may be mentioned that typically norm convergence is seen to be more prevalent than almost everywhere convergence. Various counterexamples are discussed to illustrate the need for some of the conditions in the hypotheses.
scientific article; zbMATH DE number 622557