{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:46:47Z","timestamp":1760237207574,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,3,24]],"date-time":"2020-03-24T00:00:00Z","timestamp":1585008000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"We call a subset M of an algebra of sets A a Grothendieck set for the Banach space b a ( A ) of bounded finitely additive scalar-valued measures on A equipped with the variation norm if each sequence \u03bc n n = 1 \u221e in b a ( A ) which is pointwise convergent on M is weakly convergent in b a ( A ) , i.e., if there is \u03bc \u2208 b a A such that \u03bc n A \u2192 \u03bc A for every A \u2208 M then \u03bc n \u2192 \u03bc weakly in b a ( A ) . A subset M of an algebra of sets A is called a Nikod\u00fdm set for b a ( A ) if each sequence \u03bc n n = 1 \u221e in b a ( A ) which is pointwise bounded on M is bounded in b a ( A ) . We prove that if \u03a3 is a \u03c3 -algebra of subsets of a set \u03a9 which is covered by an increasing sequence \u03a3 n : n \u2208 N of subsets of \u03a3 there exists p \u2208 N such that \u03a3 p is a Grothendieck set for b a ( A ) . This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a \u03c3 -algebra \u03a3 is covered by an increasing sequence \u03a3 n : n \u2208 N of subsets, there is p \u2208 N such that \u03a3 p is a Nikod\u00fdm set for b a \u03a3 . This also refines the Grothendieck result stating that for each \u03c3 -algebra \u03a3 the Banach space \u2113 \u221e \u03a3 is a Grothendieck space. Some applications to classic Banach space theory are given.<\/jats:p>","DOI":"10.3390\/axioms9010034","type":"journal-article","created":{"date-parts":[[2020,3,24]],"date-time":"2020-03-24T13:04:04Z","timestamp":1585055044000},"page":"34","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["On Grothendieck Sets"],"prefix":"10.3390","volume":"9","author":[{"given":"Juan Carlos","family":"Ferrando","sequence":"first","affiliation":[{"name":"Centro de Investigaci\u00f3n Operativa, Universidad Miguel Hern\u00e1ndez, E-03202 Elche, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Salvador","family":"L\u00f3pez-Alfonso","sequence":"additional","affiliation":[{"name":"Depto. Construcciones Arquitect\u00f3nicas, Universitat Polit\u00e8cnica de Val\u00e8ncia, E-46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Manuel","family":"L\u00f3pez-Pellicer","sequence":"additional","affiliation":[{"name":"Depto. de Matem\u00e1tica Aplicada and IMPA, Universitat Polit\u00e8cnica de Val\u00e8ncia, E-46022 Valencia, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,3,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"39","DOI":"10.5802\/aif.752","article-title":"On certain barrelled normed spaces","volume":"29","author":"Valdivia","year":"1979","journal-title":"Ann. Inst. Fourier (Grenoble)"},{"unstructured":"Ferrando, J.C., L\u00f3pez-Pellicer, M., and S\u00e1nchez Ruiz, L.M. (1995). Metrizable Barrelled Spaces, John Wiley & Sons Inc.. 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Convergence and boundedness of measures in non \u03c3-complete algebras, Unpublished.","key":"ref_6"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1216\/rmjm\/1181069896","article-title":"A survey on recent advances on the Nikod\u00fdm boundedness theorem and spaces of simple functions","volume":"34","author":"Ferrando","year":"2004","journal-title":"Rocky Mount. J. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/s13398-012-0081-7","article-title":"On Nikod\u00fdm boundedness property","volume":"107","author":"Valdivia","year":"2013","journal-title":"RACSAM. Rev. R. Acad. Cienc. Exactas F\u00eds. Nat. Ser. A Mat."},{"doi-asserted-by":"crossref","unstructured":"Groenewegen, G.L.M., and Van Rooij, A.C.M. (2016). Spaces of Continuous Functions, Atlantis Press. Atlantis Studies in Mathematics 4.","key":"ref_9","DOI":"10.2991\/978-94-6239-201-4"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2612","DOI":"10.1002\/mana.201600406","article-title":"On spaces Cb(X) weakly K-analytic","volume":"290","author":"Ferrando","year":"2017","journal-title":"Math. Nachr."},{"key":"ref_11","first-page":"999","article-title":"Weak convergence of bounded sequences","volume":"14","author":"Rainwater","year":"1963","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"703","DOI":"10.2140\/pjm.1972.40.703","article-title":"A convergence theorem with boundary","volume":"40","author":"Simons","year":"1972","journal-title":"Pac. J. Math."},{"key":"ref_13","first-page":"499","article-title":"Characterizations of the space of continuous functions over a compact Hausdorff space","volume":"62","author":"Arens","year":"1947","journal-title":"Trans. Am. Math. Soc."},{"unstructured":"Dunford, N., and Schwartz, J.T. (1988). Linear Operators. Part I: General Theory, John Wiley and Sons Inc.","key":"ref_14"},{"doi-asserted-by":"crossref","unstructured":"Diestel, J., and Uhl, J.J. (1977). Vector Measures, AMS. Math. Surveys and Monographs 15.","key":"ref_15","DOI":"10.1090\/surv\/015"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1017\/S0017089500030548","article-title":"Barrelled subspaces of spaces with subseries decompositions of Boolean rings of projections","volume":"36","author":"Drewnowski","year":"1994","journal-title":"Glasgow Math. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/BF01419586","article-title":"Nuclear and product spaces, Baire-like spaces and the strongest locally convex topology","volume":"197","author":"Saxon","year":"1972","journal-title":"Math. Ann."},{"doi-asserted-by":"crossref","unstructured":"Diestel, J. (1984). Sequences and Series in Banach Spaces, Springer. GTU 92.","key":"ref_18","DOI":"10.1007\/978-1-4612-5200-9"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/1\/34\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:11:13Z","timestamp":1760173873000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/9\/1\/34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3,24]]},"references-count":18,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["axioms9010034"],"URL":"https:\/\/doi.org\/10.3390\/axioms9010034","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2020,3,24]]}}}