Weight Ergodic Theorems for Probability Measures on Locally Compact Groups

Let $G$ be a locally compact group with the left Haar measure $m_G$. A probability measure $\mu$ on $G$ is said to be strictly aperiodic if the support of $\mu$ is not contained in a proper closed left coset of $G$. In this paper, we prove weight ergodic theorems for strictly aperiodic measures.