{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,20]],"date-time":"2025-06-20T19:40:10Z","timestamp":1750448410109,"version":"3.41.0"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2004,11,16]],"date-time":"2004-11-16T00:00:00Z","timestamp":1100563200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2004,12]]},"abstract":"<jats:p>We show that a measurement <jats:inline-formula>$\\mu$<\/jats:inline-formula> on a continuous dcpo <jats:inline-formula>$D$<\/jats:inline-formula> extends to a measurement <jats:inline-formula>$\\skew3\\bar{\\mu}$<\/jats:inline-formula> on the convex powerdomain <jats:inline-formula>${\\mathbf C} D$<\/jats:inline-formula> iff it is a Lebesgue measurement. In particular, <jats:inline-formula>$\\ker\\mu$<\/jats:inline-formula> must be metrisable in its relative Scott topology. Moreover, the space <jats:inline-formula>$\\ker\\skew3\\bar{\\mu}$<\/jats:inline-formula> in its relative Scott topology is homeomorphic to the Vietoris hyperspace of <jats:inline-formula>$\\ker\\mu$<\/jats:inline-formula>, that is, the space of non-empty compact subsets of <jats:inline-formula>$\\ker\\mu$<\/jats:inline-formula> in its Vietoris topology \u2013 the topology induced by any Hausdorff metric. This enables one to show that Hutchinson's theorem holds for any finite set of contractions on a domain with a Lebesgue measurement. Finally, after resolving the existence question for Lebesgue measurements on countably based domains, we uncover the following relationship between classical analysis and domain theory: for an <jats:inline-formula>$\\omega$<\/jats:inline-formula>-continuous dcpo <jats:inline-formula>$D$<\/jats:inline-formula> with <jats:inline-formula>$\\max(D)$<\/jats:inline-formula> regular, the Vietoris hyperspace of <jats:inline-formula>$\\max(D)$<\/jats:inline-formula> embeds in <jats:inline-formula>$\\max({\\mathbf C} D)$<\/jats:inline-formula> as the kernel of a measurement on <jats:inline-formula>${\\mathbf C} D$<\/jats:inline-formula>.<\/jats:p>","DOI":"10.1017\/s0960129504004384","type":"journal-article","created":{"date-parts":[[2004,11,16]],"date-time":"2004-11-16T15:28:09Z","timestamp":1100618889000},"page":"833-851","source":"Crossref","is-referenced-by-count":2,"title":["Fractals and domain theory"],"prefix":"10.1017","volume":"14","author":[{"given":"KEYE","family":"MARTIN","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2004,11,16]]},"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129504004384","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,20]],"date-time":"2025-06-20T19:13:30Z","timestamp":1750446810000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129504004384\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,11,16]]},"references-count":0,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2004,12]]}},"alternative-id":["S0960129504004384"],"URL":"https:\/\/doi.org\/10.1017\/s0960129504004384","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"type":"print","value":"0960-1295"},{"type":"electronic","value":"1469-8072"}],"subject":[],"published":{"date-parts":[[2004,11,16]]}}}