Continuity of Translation Operators

TL;DR

This paper characterizes when the space of essentially bounded functions with respect to a Radon measure remains invariant under translation operators, linking this invariance to the measure's equivalence to Lebesgue measure and providing conditions for other L^p spaces.

Contribution

It establishes necessary and sufficient conditions for translation invariance of L^p spaces under Radon measures, extending classical results to more general measures.

Findings

L^{ ext{infty}}( ext{measure}) is invariant iff measure is equivalent to Lebesgue measure

Conditions for L^p( ext{measure}) invariance in terms of Radon-Nikodym derivative

Characterization of measure invariance under translation operators

Abstract

For a Radon measure $\mu$ on $\bbR,$ 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 $L^p(\mu),\1 \leq p < \infty,$ to be invariant under the group $\{T_t\}$ in terms of the Radon-Nikodym derivative w.r.t. $m$.