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Apart from establishing basic results, we introduce a natural operation called pinning on the space of limit objects and show how this operation yields a canonical cut metric approximation to a given probability distribution akin to the weak regularity lemma for graphons. We also establish the cut metric continuity of basic operations such as taking product measures.<\/jats:p>","DOI":"10.1137\/19m126548x","type":"journal-article","created":{"date-parts":[[2021,6,1]],"date-time":"2021-06-01T11:21:25Z","timestamp":1622546485000},"page":"1096-1135","source":"Crossref","is-referenced-by-count":3,"title":["The Cut Metric for Probability Distributions"],"prefix":"10.1137","volume":"35","author":[{"given":"Amin","family":"Coja-Oghlan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Max","family":"Hahn-Klimroth","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,6,1]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(81)90099-3"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-0000(03)00008-4"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1214\/08-PS124"},{"key":"atypb4","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1214\/13-AIHP584","volume":"51","author":"Austin T.","year":"2015","journal-title":"Ann. 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