{"paper":{"title":"Covering theorems and Lebesgue integration","license":"","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Erik Talvila, Peter A. Loeb","submitted_at":"2001-01-02T19:09:48Z","abstract_excerpt":"This paper shows how the Lebesgue integral can be obtained as a Riemann sum and provides an extension of the Morse Covering Theorem to open sets. Let $X$ be a finite dimensional normed space; let $\\mu$ be a Radon measure on $X$ and let $\\Omega\\subseteq X$ be a $\\mu$-measurable set. For $\\lambda\\geq1$, a $\\mu $-measurable set $S_{\\lambda}(a)\\subseteq X$ is a $\\lambda$-Morse set with tag $a\\in S_{\\lambda}(a)$ if there is $r>0$ such that $B(a,r)\\subseteq S_{\\lambda }(a)\\subseteq B(a,\\lambda r)$ and $S_{\\lambda}(a)$ is starlike with respect to all points in the closed ball $B(a,r)$. Given a gauge "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0101014","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0101014/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}