{"paper":{"title":"On the commutation of generalized means on probability spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Janusz Matkowski, Paolo Leonetti, Salvatore Tringali","submitted_at":"2015-03-03T21:50:27Z","abstract_excerpt":"Let $f$ and $g$ be real-valued continuous injections defined on a non-empty real interval $I$, and let $(X, \\mathscr{L}, \\lambda)$ and $(Y, \\mathscr{M}, \\mu)$ be probability spaces in each of which there is at least one measurable set whose measure is strictly between $0$ and $1$.\n  We say that $(f,g)$ is a $(\\lambda, \\mu)$-switch if, for every $\\mathscr{L} \\otimes \\mathscr{M}$-measurable function $h: X \\times Y \\to \\mathbf{R}$ for which $h[X\\times Y]$ is contained in a compact subset of $I$, it holds $$ f^{-1}\\!\\left(\\int_X f\\!\\left(g^{-1}\\!\\left(\\int_Y g \\circ h\\;d\\mu\\right)\\right)d \\lambda\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01139","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"}