Continuity of Translation Operators

Abstract

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 equivalent to Lebesgue measure $m$. We also give necessary and sufficient conditions for $L^p(μ),\1 \leq p < \infty,$ to be invariant under the group $\{T_t\}$ in terms of the Radon-Nikodym derivative w.r.t. $m$.

to appear, PAMS