{"paper":{"title":"On natural density, orthomodular lattices, measure algebras and non-distributive $L^p$ spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN","math.NT","math.QA"],"primary_cat":"math.FA","authors_text":"Jarno Talponen","submitted_at":"2015-01-03T19:46:44Z","abstract_excerpt":"In this note we show, roughly speaking, that if $\\mathcal{B}$ is a Boolean algebra included in the natural way in the collection $\\mathcal{D}/_\\sim$ of all equivalence classes of natural density sets of the natural numbers, modulo null density, then $\\mathcal{B}$ extends to a $\\sigma$-algebra $\\Sigma \\subset \\mathcal{D}/_\\sim$ and the natural density is $\\sigma$-additive on $\\Sigma$. We prove the main tool employed in the argument in a more general setting, involving a kind of quantum state function, more precisely, a group-valued submeasure on an orthomodular lattice. At the end we discuss th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00597","kind":"arxiv","version":3},"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"}