Article
MSC:
06F20,
28A12,
28A33,
46A55,
46B42 | MR 3390280 | Zbl 06486997 | DOI: 10.14712/1213-7243.2015.130
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Keywords:
linear lattice; order bounded; additive set function; quasi-measure; atomic; extension; convex set; extreme point; weakly compact
linear lattice; order bounded; additive set function; quasi-measure; atomic; extension; convex set; extreme point; weakly compact
Summary:
Let $\mathfrak M$ and $\mathfrak R$ be algebras of subsets of a set $\Omega $ with $\mathfrak M\subset\mathfrak R$, and denote by $E(\mu )$ the set of all quasi-measure extensions of a given quasi-measure $\mu $ on $\mathfrak M$ to $\mathfrak R$. We give some criteria for order boundedness of $E(\mu )$ in $ba(\mathfrak R)$, in the general case as well as for atomic $\mu $. Order boundedness implies weak compactness of $E (\mu )$. We show that the converse implication holds under some assumptions on $\mathfrak M$, $\mathfrak R$ and $\mu $ or $\mu $ alone, but not in general.
References:
[1] Aliprantis C.D., Burkinshaw O.: Positive Operators. Academic Press, Orlando, 1985. MR 0809372 | Zbl 1098.47001
[2] Alvarez de Araya J.: A Radon–Nikodym theorem for vector and operator valued measures. Pacific J. Math. 29 (1969), 1–10. DOI 10.2140/pjm.1969.29.1 | MR 0245753
[3] Bhaskara Rao K. P. S., Bhaskara Rao M.: Theory of Charges. A Study of Finitely Additive Measures. Academic Press, London, 1983. MR 0751777 | Zbl 0516.28001
[4] Bourbaki N.: Espaces vectoriels topologiques. Chapitres 1 à 5, Springer, Berlin, 2007. Zbl 1106.46003
[5] Brooks J. K.: Weak compactness in the space of vector measures. Bull. Amer. Math. Soc. 78 (1972), 284–287. DOI 10.1090/S0002-9904-1972-12960-4 | MR 0324408 | Zbl 0241.28011
[6] Drewnowski L.: Topological rings of sets, continuous set functions, integration. I. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 269–276. MR 0306432 | Zbl 0249.28004
[7] Hoffmann-Jørgensen J.: Vector measures. Math. Scand. 28 (1971), 5–32. MR 0306438 | Zbl 0217.38001
[8] Lipecki Z.: On compactness and extreme points of some sets of quasi-measures and measures. Manuscripta Math. 86 (1995), 349–365. DOI 10.1007/BF02567999 | MR 1323797 | Zbl 1118.28003
[9] Lipecki Z.: On compactness and extreme points of some sets of quasi-measures and measures. II. Manuscripta Math. 89 (1996), 395–406. DOI 10.1007/BF02567525 | MR 1378601 | Zbl 0847.28001
[10] Lipecki Z.: Quasi-measures with finitely or countably many extreme extensions. Manuscripta Math. 97 (1998), 469–481. DOI 10.1007/s002290050115 | MR 1660148 | Zbl 0918.28003
[11] Lipecki Z.: Cardinality of the set of extreme extensions of a quasi-measure. Manuscripta Math. 104 (2001), 333–341. DOI 10.1007/s002290170031 | MR 1828879 | Zbl 1041.28001
[12] Lipecki Z.: The variation of an additive function on a Boolean algebra. Publ. Math. Debrecen 63 (2003), 445–459. MR 2018076 | Zbl 1064.28005
[13] Lipecki Z.: On compactness and extreme points of some sets of quasi-measures and measures. III. Manuscripta Math. 117 (2005), 463–473. DOI 10.1007/s00229-005-0571-4 | MR 2163488 | Zbl 1092.28002
[14] Lipecki Z.: On compactness and extreme points of some sets of quasi-measures and measures. IV. Manuscripta Math. 123 (2007), 133–146. DOI 10.1007/s00229-007-0085-3 | MR 2306629 | Zbl 1118.28003
[15] Lipecki Z.: Cardinality of some convex sets and of their sets of extreme points. Colloq. Math. 123 (2011), 133–147. DOI 10.4064/cm123-1-10 | MR 2794124 | Zbl 1223.28002
[16] Lipecki Z.: Compactness and extreme points of the set of quasi-measure extensions of a quasi-measure. Dissertationes Math. (Rozprawy Mat.) 493 (2013), 59 pp. MR 3135305 | Zbl 1283.28002
[17] Marczewski E.: Measures in almost independent fields. Fund. Math. 38 (1951), 217–229; reprinted in: Marczewski E., Collected Mathematical Papers, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 1996, pp. 413–425. MR 0047116 | Zbl 0045.02303
[18] Nygaard O., P oldvere M.: Families of vector measures of uniformly bounded variation. Arch. Math. (Basel) 88 (2007), 57–61. DOI 10.1007/s00013-006-1859-7 | MR 2289601
[19] Plachky D.: Extremal and monogenic additive set functions. Proc. Amer. Math. Soc. 54 (1976), 193–196. DOI 10.1090/S0002-9939-1976-0419711-3 | MR 0419711 | Zbl 0285.28005
[20] Wnuk W.: Banach Lattices with Order Continuous Norms. PWN---Polish Sci. Publ., Warszawa, 1999. Zbl 0948.46017
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