The Ergodicity of the Convolution $\mu \ast \upsilon$ on a Vector Space

Abstract

Let $\mathrm{G}$ be a subgroup of a vector space $X$ and $\mu ,\upsilon$ be two probability measures on $X$. If $\mu$ and $\upsilon$ are $\mathrm{G}$-quasi-invariant and $\mathrm{G}$-ergodic, then the convolution $\mu \ast \upsilon$ is also $\mathrm{G}$-ergodic.