{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:50:45Z","timestamp":1760403045871,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,4,23]],"date-time":"2020-04-23T00:00:00Z","timestamp":1587600000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005357","name":"Slovak Research and Development Agency","doi-asserted-by":"publisher","award":["no.\\ APVV-17-0066 and no.\\ APVV-18-0052."],"award-info":[{"award-number":["no.\\ APVV-17-0066 and no.\\ APVV-18-0052."]}],"id":[{"id":"10.13039\/501100005357","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet integrals, do not cover this important class of functionals on real random variables. In this paper, a new approach to the construction of coherent lower previsions acting on a finite space is proposed, exemplified and studied. It is based on special decomposition integrals recently introduced by Even and Lehrer, in our case the considered decomposition systems being single collections and thus called collection integrals. In special case when these integrals, defined for non-negative random variables only, are shift-invariant, we extend them to the class of all real random variables, thus obtaining so called super-additive integrals. Our proposed construction can be seen then as a normalized super-additive integral. We discuss and exemplify several particular cases, for example, when collections determine a coherent lower prevision for any monotone set function. For some particular collections, only particular set functions can be considered for our construction. Conjugated coherent upper previsions are also considered.<\/jats:p>","DOI":"10.3390\/axioms9020043","type":"journal-article","created":{"date-parts":[[2020,4,23]],"date-time":"2020-04-23T10:46:22Z","timestamp":1587638782000},"page":"43","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Integral Representation of Coherent Lower Previsions by Super-Additive Integrals"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9517-8407","authenticated-orcid":false,"given":"Serena","family":"Doria","sequence":"first","affiliation":[{"name":"Department of Engineering and Geology, University G. d\u2019Annunzio, 66100 Chieti-Pescara, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6503-949X","authenticated-orcid":false,"given":"Radko","family":"Mesiar","sequence":"additional","affiliation":[{"name":"Faculty of Civil Engineering, Slovak University of Technology, Radlinsk\u00e9ho 11, 810 05 Bratislava, Slovakia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3215-6195","authenticated-orcid":false,"given":"Adam","family":"\u0160eliga","sequence":"additional","affiliation":[{"name":"Faculty of Civil Engineering, Slovak University of Technology, Radlinsk\u00e9ho 11, 810 05 Bratislava, Slovakia"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities, Chapman and Hall.","DOI":"10.1007\/978-1-4899-3472-7"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Doria, S., Mesiar, R., and \u0160eliga, A. (2020). Construction method of coherent lower and upper previsions based on collection integrals. Boll. Dell\u2019Unione Mat. Ital.","DOI":"10.1007\/s40574-020-00220-1"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Denneberg, D. (1994). Non-Additive Measure and Integral, Kluwer Academic.","DOI":"10.1007\/978-94-017-2434-0"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"131","DOI":"10.5802\/aif.53","article-title":"Theory of capacities","volume":"5","author":"Choquet","year":"1954","journal-title":"Ann. L\u2019Institut Fourier"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/S1385-7258(71)80017-3","article-title":"Maxitive measure and integration","volume":"33","author":"Shilkret","year":"1971","journal-title":"Indag. Math."},{"key":"ref_6","unstructured":"Sugeno, M. (1974). Theory of Fuzzy Integrals and Its Applications. [Ph.D. Thesis, Tokyo Institute of Technology]."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s00199-013-0780-0","article-title":"Decomposition-integral: Unifying Choquet and the concave integrals","volume":"56","author":"Even","year":"2014","journal-title":"Econ. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1252","DOI":"10.1016\/j.ijar.2013.02.001","article-title":"Decomposition integrals","volume":"54","author":"Mesiar","year":"2013","journal-title":"Int. J. Approx. Reason."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Wang, Z., and Klir, G.J. (2009). Generalized Measure Theory, Springer.","DOI":"10.1007\/978-0-387-76852-6"},{"key":"ref_10","first-page":"107","article-title":"The PAN-integral on the fuzzy measure space","volume":"3","author":"Yang","year":"1985","journal-title":"Fuzzy Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/s00199-007-0302-z","article-title":"A new integral for capacities","volume":"39","author":"Lehrer","year":"2009","journal-title":"Econ. Theory"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1248","DOI":"10.1016\/j.ijar.2012.06.018","article-title":"Forecasting with imprecise probabilities","volume":"53","author":"Seidenfeld","year":"2012","journal-title":"Int. J. Approx. Reason."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Troffaes, M., and De Cooman, G. (2014). Lower Previsions, Wiley.","DOI":"10.1002\/9781118762622"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1080\/03081079.2018.1473392","article-title":"Integral representation of coherent upper conditional prevision with respect to its associated Hausdorff outer measure: A comparison among the Choquet integral, the pan-integral and the concave integral","volume":"216","author":"Doria","year":"2018","journal-title":"Int. J. Gen. Syst."},{"key":"ref_15","first-page":"41","article-title":"Decomposition integral without alternatives, its equivalence to Lebesgue integral, and computational algorithms","volume":"13","year":"2019","journal-title":"J. Autom. Mob. Robot. Intell. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"845","DOI":"10.1214\/aos\/1176350379","article-title":"De Finetti\u2019s coherence and statistical inference","volume":"15","author":"Regazzini","year":"1987","journal-title":"Ann. Stat."},{"key":"ref_17","unstructured":"De Finetti, B. (1974). Theory of Probability, Wiley."},{"key":"ref_18","unstructured":"De Finetti, B. (1972). Probability, Induction and Statistics, Wiley."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1214\/aop\/1176996451","article-title":"Finitely additive conditional probabilities, conglomerability and disintegrations","volume":"3","author":"Dubins","year":"1975","journal-title":"Ann. Probab."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/BF02924866","article-title":"Finitely additive conditional probabilities","volume":"55","author":"Regazzini","year":"1985","journal-title":"Rendiconti del Seminario Matematico e Fisico di Milano"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"391","DOI":"10.2307\/2025161","article-title":"On indeterminate probabilities","volume":"71","author":"Levi","year":"1974","journal-title":"J. Philos."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1016\/j.dam.2019.01.024","article-title":"The cone of supermodular games on finite distributive lattices","volume":"260","author":"Grabisch","year":"2019","journal-title":"Discret. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.fss.2014.05.003","article-title":"Superdecomposition integrals","volume":"259","author":"Mesiar","year":"2015","journal-title":"Fuzzy Sets Syst."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/2\/43\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T14:27:03Z","timestamp":1760365623000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/2\/43"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,23]]},"references-count":23,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["axioms9020043"],"URL":"https:\/\/doi.org\/10.3390\/axioms9020043","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2020,4,23]]}}}