{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T14:17:42Z","timestamp":1778077062618,"version":"3.51.4"},"reference-count":57,"publisher":"Elsevier BV","issue":"8","license":[{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Annals of Pure and Applied Logic"],"published-print":{"date-parts":[[2026,8]]},"DOI":"10.1016\/j.apal.2026.103737","type":"journal-article","created":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T16:29:02Z","timestamp":1773073742000},"page":"103737","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"title":["Algorithmic randomness and the weak merging of computable probability measures"],"prefix":"10.1016","volume":"177","author":[{"given":"Simon M.","family":"Huttegger","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-5460-778X","authenticated-orcid":false,"given":"Sean","family":"Walsh","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesca","family":"Zaffora Blando","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"78","reference":[{"issue":"3","key":"10.1016\/j.apal.2026.103737_br0030","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.apal.2009.01.002","article-title":"Constructive equivalence relations on computable probability measures","volume":"160","author":"Bienvenu","year":"2009","journal-title":"Ann. Pure Appl. Log."},{"key":"10.1016\/j.apal.2026.103737_br0020","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.ic.2011.10.006","article-title":"A constructive version of Birkhoff's ergodic theorem for Martin-L\u00f6f random points","volume":"210","author":"Bienvenu","year":"2012","journal-title":"Inf. Comput."},{"issue":"3","key":"10.1016\/j.apal.2026.103737_br0010","doi-asserted-by":"crossref","first-page":"882","DOI":"10.1214\/aoms\/1177704456","article-title":"Merging of opinions with increasing information","volume":"33","author":"Blackwell","year":"1962","journal-title":"Ann. Math. Stat."},{"key":"10.1016\/j.apal.2026.103737_br0040","series-title":"Logical Foundations of Probability","author":"Carnap","year":"1950"},{"key":"10.1016\/j.apal.2026.103737_br0050","series-title":"Studies in Inductive Logic and Probability","first-page":"33","article-title":"A basic system of inductive logic, part 1","author":"Carnap","year":"1971"},{"key":"10.1016\/j.apal.2026.103737_br0060","series-title":"Studies in Inductive Logic and Probability","first-page":"7","article-title":"A basic system of inductive logic, part 2","author":"Carnap","year":"1980"},{"key":"10.1016\/j.apal.2026.103737_br0080","first-page":"229","article-title":"Information-type measures of difference of probability distributions and indirect observation","volume":"2","author":"Csisz\u00e1r","year":"1967","journal-title":"Studia Sci. Math. Hung."},{"issue":"4","key":"10.1016\/j.apal.2026.103737_br0070","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1561\/0100000004","article-title":"Information theory and statistics: a tutorial","volume":"1","author":"Csisz\u00e1r","year":"2004","journal-title":"Found. Trends Commun. Inf. Theory"},{"key":"10.1016\/j.apal.2026.103737_br0090","series-title":"Algorithmic Randomness and Complexity","author":"Downey","year":"2010"},{"key":"10.1016\/j.apal.2026.103737_br0560","series-title":"Reverse Mathematics: Problems, Reductions, and Proofs","author":"Dzhafarov","year":"2022"},{"key":"10.1016\/j.apal.2026.103737_br0110","doi-asserted-by":"crossref","first-page":"1357","DOI":"10.1093\/jrsssb\/qkad005","article-title":"Martingale posterior distributions","volume":"85","author":"Fong","year":"2023","journal-title":"J. R. Stat. Soc., Ser. B Stat. Methodol."},{"key":"10.1016\/j.apal.2026.103737_br0100","doi-asserted-by":"crossref","first-page":"3623","DOI":"10.1090\/S0002-9939-2012-11179-7","article-title":"Martin-L\u00f6f random points satisfy Birkhoff's ergodic theorem for effectively closed sets","volume":"140","author":"Franklin","year":"2012","journal-title":"Proc. Am. Math. Soc."},{"key":"10.1016\/j.apal.2026.103737_br0120","doi-asserted-by":"crossref","first-page":"711","DOI":"10.17323\/1609-4514-2014-14-4-711-744","article-title":"Randomness and non-ergodic systems","volume":"14","author":"Franklin","year":"2014","journal-title":"Mosc. Math. J."},{"issue":"1","key":"10.1016\/j.apal.2026.103737_br0570","doi-asserted-by":"crossref","first-page":"45","DOI":"10.3233\/COM-14025","article-title":"Algorithmic aspects of Lipschitz functions","volume":"3","author":"Freer","year":"2014","journal-title":"Computability"},{"issue":"3","key":"10.1016\/j.apal.2026.103737_br0130","doi-asserted-by":"crossref","first-page":"1252","DOI":"10.1109\/TIT.2007.915700","article-title":"Randomness criteria in terms of \u03b1-divergences","volume":"54","author":"Fujiwara","year":"2008","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"10.1016\/j.apal.2026.103737_br0140","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1111\/j.1751-5823.2002.tb00178.x","article-title":"On choosing and bounding probability metrics","volume":"70","author":"Gibbs","year":"2002","journal-title":"Int. Stat. Rev."},{"key":"10.1016\/j.apal.2026.103737_br0160","author":"Hoyrup"},{"key":"10.1016\/j.apal.2026.103737_br0170","series-title":"Handbook of Computability and Complexity in Analysis","first-page":"227","article-title":"Computable measure theory and algorithmic randomness","author":"Hoyrup","year":"2021"},{"key":"10.1016\/j.apal.2026.103737_br0180","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1017\/S1755020315000180","article-title":"Merging of opinions and probability kinematics","volume":"8","author":"Huttegger","year":"2015","journal-title":"Rev. Symb. Log."},{"key":"10.1016\/j.apal.2026.103737_br0190","series-title":"The Probabilistic Foundations of Rational Learning","author":"Huttegger","year":"2017"},{"key":"10.1016\/j.apal.2026.103737_br0200","author":"Huttegger"},{"key":"10.1016\/j.apal.2026.103737_br0150","series-title":"Algorithmic Learning Theory","first-page":"234","article-title":"Universal convergence of semimeasures on individual random sequences","author":"Hutter","year":"2004"},{"key":"10.1016\/j.apal.2026.103737_br0420","unstructured":"Jason Rute, Algorithmic randomness, martingales, and differentiability I, preprint, 2012."},{"issue":"2(10)","key":"10.1016\/j.apal.2026.103737_br0260","first-page":"227","article-title":"On the question of the absolute continuity and singularity of probability measures","volume":"104(146)","author":"Kabanov","year":"1977","journal-title":"Mat. Sb. Nov. Ser."},{"issue":"1","key":"10.1016\/j.apal.2026.103737_br0210","doi-asserted-by":"crossref","first-page":"214","DOI":"10.2307\/1969123","article-title":"On equivalence of infinite product measures","volume":"49","author":"Kakutani","year":"1948","journal-title":"Ann. Math."},{"key":"10.1016\/j.apal.2026.103737_br0220","series-title":"Merging economic forecasts","author":"Kalai","year":"1990"},{"key":"10.1016\/j.apal.2026.103737_br0230","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.2307\/2951492","article-title":"Rational learning leads to Nash equilibrium","volume":"61","author":"Kalai","year":"1993","journal-title":"Econometrica"},{"key":"10.1016\/j.apal.2026.103737_br0240","doi-asserted-by":"crossref","first-page":"1231","DOI":"10.2307\/2951500","article-title":"Subjective equilibrium in repeated games","volume":"61","author":"Kalai","year":"1993","journal-title":"Econometrica"},{"issue":"1","key":"10.1016\/j.apal.2026.103737_br0250","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0304-4068(94)90037-X","article-title":"Weak and strong merging of opinions","volume":"23","author":"Kalai","year":"1994","journal-title":"J. Math. Econ."},{"key":"10.1016\/j.apal.2026.103737_br0270","series-title":"Recursion Theory Week","first-page":"245","article-title":"Measure \u03a010-classes and complete extensions of PA","volume":"vol. 1141","author":"Kucera","year":"1985"},{"issue":"2","key":"10.1016\/j.apal.2026.103737_br0280","first-page":"337","article-title":"Uniform tests of randomness","volume":"17","author":"Levin","year":"1976","journal-title":"Sov. Math. Dokl."},{"key":"10.1016\/j.apal.2026.103737_br0290","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-11298-1","article-title":"An Introduction to Kolmogorov Complexity and Its Applications","author":"Li","year":"2019"},{"key":"10.1016\/j.apal.2026.103737_br0320","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1016\/S0019-9958(66)80018-9","article-title":"The definition of random sequences","volume":"9","author":"Martin-L\u00f6f","year":"1966","journal-title":"Inf. Control"},{"issue":"1","key":"10.1016\/j.apal.2026.103737_br0300","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/s00224-012-9422-3","article-title":"Characterization of Kurtz randomness by a differentiation theorem","volume":"52","author":"Miyabe","year":"2013","journal-title":"Theory Comput. Syst."},{"key":"10.1016\/j.apal.2026.103737_br0310","doi-asserted-by":"crossref","first-page":"15","DOI":"10.3233\/COM-13015","article-title":"L1-computability, layerwise computability, and Solovay reducibility","volume":"2","author":"Miyabe","year":"2013","journal-title":"Computability"},{"key":"10.1016\/j.apal.2026.103737_br0330","series-title":"Probabilistic Machine Learning: Advanced Topics","author":"Murphy","year":"2022"},{"key":"10.1016\/j.apal.2026.103737_br0340","doi-asserted-by":"crossref","first-page":"275","DOI":"10.2307\/2171894","article-title":"Prediction, optimization, and learning in repeated games","volume":"65","author":"Nachbar","year":"1997","journal-title":"Econometrica"},{"key":"10.1016\/j.apal.2026.103737_br0350","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1111\/j.1468-0262.2005.00585.x","article-title":"Beliefs in repeated games","volume":"73","author":"Nachbar","year":"2005","journal-title":"Econometrica"},{"key":"10.1016\/j.apal.2026.103737_br0360","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.geb.2022.09.002","article-title":"The possibility of Bayesian learning in repeated games","volume":"136","author":"Norman","year":"2022","journal-title":"Games Econ. Behav."},{"issue":"6","key":"10.1016\/j.apal.2026.103737_br0370","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1007\/BF01213814","article-title":"Bayesian learning leads to correlated equilibria in normal form games","volume":"4","author":"Nyarko","year":"1994","journal-title":"Econ. Theory"},{"key":"10.1016\/j.apal.2026.103737_br0380","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1007\/s001990050206","article-title":"Bayesian learning and convergence to Nash equilibria without common priors","volume":"11","author":"Nyarko","year":"1998","journal-title":"Econ. Theory"},{"issue":"1","key":"10.1016\/j.apal.2026.103737_br0400","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1090\/S0002-9939-2013-11710-7","article-title":"Schnorr randomness and the Lebesgue differentiation theorem","volume":"142","author":"Pathak","year":"2014","journal-title":"Proc. Am. Math. Soc."},{"key":"10.1016\/j.apal.2026.103737_br0390","series-title":"A User's Guide to Measure Theoretic Probability","author":"Pollard","year":"2002"},{"key":"10.1016\/j.apal.2026.103737_br0410","series-title":"Information Theory: From Coding to Learning","author":"Polyanskiy","year":"2025"},{"issue":"1","key":"10.1016\/j.apal.2026.103737_br0430","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1006\/jeth.1997.2352","article-title":"Does rational learning lead to Nash equilibrium in finitely repeated games?","volume":"78","author":"Sandroni","year":"1998","journal-title":"J. Econ. Theory"},{"issue":"1","key":"10.1016\/j.apal.2026.103737_br0440","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1006\/game.1997.0572","article-title":"Necessary and sufficient conditions for convergence to Nash equilibrium: the almost absolute continuity hypothesis","volume":"22","author":"Sandroni","year":"1998","journal-title":"Games Econ. Behav."},{"key":"10.1016\/j.apal.2026.103737_br0450","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1007\/BF01694181","article-title":"A unified approach to the definition of a random sequence","volume":"5","author":"Schnorr","year":"1971","journal-title":"Math. Syst. Theory"},{"key":"10.1016\/j.apal.2026.103737_br0460","article-title":"Zuf\u00e4lligkeit und Wahrscheinlichkeit. Eine algorithmische Begr\u00fcndung der Wahrscheinlichkeitstheorie","volume":"vol. 218","author":"Schnorr","year":"1971"},{"key":"10.1016\/j.apal.2026.103737_br0470","series-title":"Probability-2","author":"Shiryaev","year":"2019"},{"key":"10.1016\/j.apal.2026.103737_br0480","series-title":"Subsystems of Second Order Arithmetic","author":"Simpson","year":"2009"},{"key":"10.1016\/j.apal.2026.103737_br0490","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0019-9958(64)90223-2","article-title":"A formal theory of inductive inference: parts 1 and 2","volume":"7","author":"Solomonoff","year":"1964","journal-title":"Inf. Control"},{"issue":"4","key":"10.1016\/j.apal.2026.103737_br0500","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1109\/TIT.1978.1055913","article-title":"Complexity-based induction systems: comparisons and convergence theorems","volume":"24","author":"Solomonoff","year":"1978","journal-title":"IEEE Trans. Inf. Theory"},{"key":"10.1016\/j.apal.2026.103737_br0510","series-title":"Algorithmic Randomness: Progress and Prospects","first-page":"40","article-title":"Algorithmic randomness and ergodic theory","author":"Towsner","year":"2020"},{"issue":"6","key":"10.1016\/j.apal.2026.103737_br0520","first-page":"1298","article-title":"On a criterion for randomness","volume":"294","author":"Vovk","year":"1987","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"10.1016\/j.apal.2026.103737_br0530","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1007\/s10463-007-0165-x","article-title":"Merging of opinions in game-theoretic probability","volume":"61","author":"Vovk","year":"2009","journal-title":"Ann. Inst. Stat. Math."},{"key":"10.1016\/j.apal.2026.103737_br0540","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511813658","article-title":"Probability with Martingales","author":"Williams","year":"1991"},{"key":"10.1016\/j.apal.2026.103737_br0550","article-title":"Bayesian merging of opinions and algorithmic randomness","author":"Zaffora Blando","year":"2022","journal-title":"Br. J. Philos. Sci."}],"container-title":["Annals of Pure and Applied Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0168007226000205?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0168007226000205?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T13:42:26Z","timestamp":1778074946000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0168007226000205"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8]]},"references-count":57,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["S0168007226000205"],"URL":"https:\/\/doi.org\/10.1016\/j.apal.2026.103737","relation":{},"ISSN":["0168-0072"],"issn-type":[{"value":"0168-0072","type":"print"}],"subject":[],"published":{"date-parts":[[2026,8]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Algorithmic randomness and the weak merging of computable probability measures","name":"articletitle","label":"Article Title"},{"value":"Annals of Pure and Applied Logic","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.apal.2026.103737","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"103737"}}