{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,18]],"date-time":"2025-12-18T19:06:44Z","timestamp":1766084804449,"version":"3.48.0"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T00:00:00Z","timestamp":1763942400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T00:00:00Z","timestamp":1763942400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Log. Univers."],"published-print":{"date-parts":[[2025,12]]},"abstract":"Abstract<\/jats:title>\n \n This paper introduces a model-theoretic generalization of the notion of forcing with random reals, in which forcing gives rise to\n random generic structures<\/jats:italic>\n . Specifically, we consider forcing with\n \n \n $$\\kappa $$<\/jats:tex-math>\n \n \u03ba<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -Borel probability measures on the space of\n \n \n $$\\mathscr {L}$$<\/jats:tex-math>\n \n L<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -structures with a (possibly uncountable) infinite set\n X<\/jats:italic>\n , focusing on those that are invariant under the action of the symmetric group\n \n \n $${{\\,\\textrm{Sym}\\,}}{(X)}$$<\/jats:tex-math>\n \n \n \n \n Sym<\/mml:mtext>\n \n <\/mml:mrow>\n \n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We demonstrate how any\n \n \n $${{\\,\\textrm{Sym}\\,}}{(X)}$$<\/jats:tex-math>\n \n \n \n \n Sym<\/mml:mtext>\n \n <\/mml:mrow>\n \n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -invariant measure where\n X<\/jats:italic>\n is countable can be uniquely extended to a\n \n \n $${{\\,\\textrm{Sym}\\,}}{(Y)}$$<\/jats:tex-math>\n \n \n \n \n Sym<\/mml:mtext>\n \n <\/mml:mrow>\n \n (<\/mml:mo>\n Y<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -invariant measure where\n Y<\/jats:italic>\n is uncountable, and prove that forcing with such measures satisfies the countable chain condition. We also show that we can uniformly distinguish between these random generic structures and the\n Cohen generic structures<\/jats:italic>\n that arise from forcing with a strong Fra\u00efss\u00e9 class: There is a\n \n \n $$\\kappa $$<\/jats:tex-math>\n \n \u03ba<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -Borel set of low complexity that contains every Cohen generic structure that is not highly homogeneous but contains no random generic structure, implying that a structure that is not highly homogeneous cannot be both Cohen generic and random generic. Finally, we answer an open question of Kostana in the case of\n \n \n $$\\omega _1$$<\/jats:tex-math>\n \n \n \u03c9<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , by establishing a connection between forcing with a strong Fra\u00efss\u00e9 class and Cohen forcing.\n <\/jats:p>","DOI":"10.1007\/s11787-025-00394-2","type":"journal-article","created":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T14:22:37Z","timestamp":1763994157000},"page":"799-840","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Forcing with Invariant Measures"],"prefix":"10.1007","volume":"19","author":[{"given":"Nathanael","family":"Ackerman","sequence":"first","affiliation":[]},{"given":"Cameron","family":"Freer","sequence":"additional","affiliation":[]},{"given":"Mohammad","family":"Golshani","sequence":"additional","affiliation":[]},{"given":"Mostafa","family":"Mirabi","sequence":"additional","affiliation":[]},{"given":"Rehana","family":"Patel","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,11,24]]},"reference":[{"issue":"10","key":"394_CR1","doi-asserted-by":"publisher","first-page":"1299","DOI":"10.1016\/j.apal.2010.04.003","volume":"161","author":"NL Ackerman","year":"2010","unstructured":"Ackerman, N.L.: Relativized Grothendieck topoi. 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