{"paper":{"title":"On Various Modes of Scalar Convergence in L_0(X)","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Maria Girardi, Stephen J. Dilworth","submitted_at":"1996-04-05T00:00:00Z","abstract_excerpt":"A sequence $\\{f_n\\}$ of strongly-measurable functions taking values in a Banach space $\\X$ is scalarly null a\\.e\\. (resp. scalarly null in measure) if $x^*f_n \\rightarrow0$ a\\.e\\. (resp. $x^*f_n \\rightarrow 0$ in measure) for every $x^*\\in \\X^*$. Let $1\\le p\\le \\infty$. The main questions addressed in this paper are whether an $L_p(\\X)$-bounded sequence that is scalarly null a\\.e\\. will converge weakly a\\.e\\. (or have a subsequence which converges weakly a\\.e\\.), and whether an $L_p(\\X)$-bounded sequence that is scalarly null in measure will have a subsequence that is scalarly null a\\.e. The a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9604212","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"}