A sequence \(\mu_ n\) of probabilities on a locally compact \(\sigma\)- compact group \(G\) is by definition ergodic if \(\| \mu_ n*(f - \delta_ g*f) \|_ 1 \to 0\) for any function \(f\), square integrable with respect to the right Haar measure. The authors give six equivalent conditions for this property. One of them is that \(\| U_{\mu_ n} (I-T(g))x \| \to 0\) \((x \in X\), \(g \in G)\), where \(T(g)\) is a bounded strongly continuous representation of \(G\) by linear operators on some Banach space \(X\), and where \(U_ \mu x = \int T(g)x \mu (dg)\) for a probability \(\mu\). In a final section the a.e. convergence of \((U_ \mu)^ nh\) is proven under suitable conditions when the representation is by Lamperti operators. This includes a generalization of Calderon’s transfer principle for maximal inequalities.