{"paper":{"title":"A note on duality theorems in mass transportation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Pietro Rigo","submitted_at":"2019-07-16T15:20:19Z","abstract_excerpt":"The duality theory of the Monge-Kantorovich transport problem is investigated in an abstract measure theoretic framework. Let $(\\mathcal{X},\\mathcal{F},\\mu)$ and $(\\mathcal{Y},\\mathcal{G},\\nu)$ be any probability spaces and $c:\\mathcal{X}\\times\\mathcal{Y}\\rightarrow\\mathbb{R}$ a measurable cost function such that $f_1+g_1\\le c\\le f_2+g_2$ for some $f_1,\\,f_2\\in L_1(\\mu)$ and $g_1,\\,g_2\\in L_1(\\nu)$. Define $\\alpha(c)=\\inf_P\\int c\\,dP$ and $\\alpha^*(c)=\\sup_P\\int c\\,dP$, where $\\inf$ and $\\sup$ are over the probabilities $P$ on $\\mathcal{F}\\otimes\\mathcal{G}$ with marginals $\\mu$ and $\\nu$. Som"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.07059","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"}