{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:53:25Z","timestamp":1753894405115,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>Bishop's measure theory (BMT) is an abstraction of the measure theory of a\nlocally compact metric space $X$, and the use of an informal notion of a\nset-indexed family of complemented subsets is crucial to its predicative\ncharacter. The more general Bishop-Cheng measure theory (BCMT) is a\nconstructive version of the classical Daniell approach to measure and\nintegration, and highly impredicative, as many of its fundamental notions, such\nas the integration space of $p$-integrable functions $L^p$, rely on\nquantification over proper classes (from the constructive point of view). In\nthis paper we introduce the notions of a pre-measure and pre-integration space,\na predicative variation of the Bishop-Cheng notion of a measure space and of an\nintegration space, respectively. Working within Bishop Set Theory (BST), and\nusing the theory of set-indexed families of complemented subsets and\nset-indexed families of real-valued partial functions within BST, we apply the\nimplicit, predicative spirit of BMT to BCMT. As a first example, we present the\npre-measure space of complemented detachable subsets of a set $X$ with the\nDirac-measure, concentrated at a single point. Furthermore, we translate in our\npredicative framework the non-trivial, Bishop-Cheng construction of an\nintegration space from a given measure space, showing that a pre-measure space\ninduces the pre-integration space of simple functions associated to it.\nFinally, a predicative construction of the canonically integrable functions\n$L^1$, as the completion of an integration space, is included.<\/jats:p>","DOI":"10.46298\/lmcs-20(4:2)2024","type":"journal-article","created":{"date-parts":[[2024,10,7]],"date-time":"2024-10-07T19:10:06Z","timestamp":1728328206000},"source":"Crossref","is-referenced-by-count":0,"title":["Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory"],"prefix":"10.46298","volume":"Volume 20, Issue 4","author":[{"given":"Iosif","family":"Petrakis","sequence":"first","affiliation":[]},{"given":"Max","family":"Zeuner","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2024,10,7]]},"container-title":["Logical Methods in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/lmcs.episciences.org\/14409\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/lmcs.episciences.org\/14409\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,7]],"date-time":"2024-10-07T19:10:07Z","timestamp":1728328207000},"score":1,"resource":{"primary":{"URL":"https:\/\/lmcs.episciences.org\/9808"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,7]]},"references-count":0,"URL":"https:\/\/doi.org\/10.46298\/lmcs-20(4:2)2024","relation":{"has-preprint":[{"id-type":"arxiv","id":"2207.08684v2","asserted-by":"subject"},{"id-type":"arxiv","id":"2207.08684v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"2207.08684","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.2207.08684","asserted-by":"subject"}]},"ISSN":["1860-5974"],"issn-type":[{"type":"electronic","value":"1860-5974"}],"subject":[],"published":{"date-parts":[[2024,10,7]]},"article-number":"9808"}}