{"paper":{"title":"Relations between $\\mathcal{L}^p$- and Pointwise Convergence of Families of Functions Indexed by the Unit Interval","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Christian M\\\"onch, Vaios Laschos","submitted_at":"2012-02-26T21:54:43Z","abstract_excerpt":"We construct a variety of mappings of the unit interval into $\\mathcal{L}^p([0,1])$ to generalize classical examples of $\\mathcal{L}^p$-convergence of sequences of functions with simultaneous pointwise divergence. By establishing relations between the regularity of the functions in the image of the mappings and the topology of $[0,1]$, we obtain examples which are $\\mathcal{L}^p$-continuous but exhibit discontinuity in a pointwise sense to different degrees. We conclude by proving an Egorov-type theorem, namely that if almost every function in the image is continuous, then we can remove a set "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.5792","kind":"arxiv","version":2},"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"}