{"paper":{"title":"Continuity of Translation Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Justin R. Peters, Krishna B. Athreya","submitted_at":"2010-10-31T21:33:15Z","abstract_excerpt":"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. $m$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.0212","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}