Continuity of Translation Operators

Krishna B. Athreya, Justin R. Peters

For a Radon measure $\mu$ on $\mathbb{R},$ we show that $L^{\infty }(\mu )$ is invariant under the group of translation operators $T_t(f)(x) = {$f(x-t)$}\ (t \in \bbR)$ if and only if $\mu$ is e…