{"paper":{"title":"Lebesgue classes and preparation of real constructible functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.LO"],"primary_cat":"math.AG","authors_text":"Daniel J. Miller, Raf Cluckers","submitted_at":"2012-09-15T21:32:29Z","abstract_excerpt":"We call a function constructible if it has a globally subanalytic domain and can be expressed as a sum of products of globally subanalytic functions and logarithms of positively-valued globally subanalytic functions. For any $q > 0$ and constructible functions $f$ and $\\mu$ on $E\\times\\RR^n$, we prove a theorem describing the structure of the set of all $(x,p)$ in $E \\times (0,\\infty]$ for which $y \\mapsto f(x,y)$ is in $L^p(|\\mu|_{x}^{q})$, where $|\\mu|_{x}^{q}$ is the positive measure on $\\RR^n$ whose Radon-Nikodym derivative with respect to the Lebesgue measure is $y\\mapsto |\\mu(x,y)|^q$. W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3439","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"}