Article
MSC:
06F20,
28A12,
28A33,
46A55,
46B42 | MR 3666941 | Zbl 06773714 | DOI: 10.14712/1213-7243.2015.208
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Keywords:
linear lattice; ideal; order bounded; ideal dominated; order unit; Banach lattice; $\textit{AM}$-space; convex set; extreme point; weakly compact; additive set function; quasi-measure; atomic; extension
linear lattice; ideal; order bounded; ideal dominated; order unit; Banach lattice; $\textit{AM}$-space; convex set; extreme point; weakly compact; additive set function; quasi-measure; atomic; extension
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 show that $E(\mu )$ is order bounded if and only if it is contained in a principal ideal in $ba(\mathfrak R)$ if and only if it is weakly compact and $\operatorname{extr} E(\mu )$ is contained in a principal ideal in $ba(\mathfrak R)$. We also establish some criteria for the coincidence of the ideals, in $ba(\mathfrak R)$, generated by $E(\mu )$ and $\operatorname{extr} E(\mu )$.
References:
[1] Abramovič Ju. A.: Some theorems on normed lattices. Vestnik Leningrad. Univ. 13 (1971), 5–11 (in Russian); English transl.: Vestnik Leningrad Univ. Math. 4 (1977), 153–159. MR 0288554
[2] Aliprantis C.D., Burkinshaw O.: Locally Solid Riesz Spaces. Academic Press, Orlando, 1978. MR 0493242 | Zbl 1043.46003
[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] 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
[5] 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
[6] 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
[7] 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
[8] 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
[9] Lipecki Z.: Order boundedness and weak compactness of the set of quasi-measure extensions of a quasi-measure. Comment. Math. Univ. Carolin. 56 (2015), 331–345. MR 3390280
[10] 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, 413–425. DOI 10.4064/fm-38-1-217-229 | MR 0047116 | Zbl 0045.02303
[11] de Pagter B., Wnuk W.: Some remarks on Banach lattices with non-atomic duals. Indag. Math. (N.S.) 1 (1990), 391–395. DOI 10.1016/0019-3577(90)90026-J | MR 1075887 | Zbl 0731.46008
[12] 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
[13] Schaefer H.H.: Topological Vector Spaces. Macmillan, New York, 1966. MR 0193469 | Zbl 0983.46002
[14] Schaefer H.H.: Banach Lattices and Positive Operators. Springer, Berlin and New York, 1974. MR 0423039 | Zbl 0296.47023
[15] Schwarz H.-U.: Banach Lattices and Operators. Teubner, Leipzig, 1984. MR 0781131 | Zbl 0585.47025
[16] Zaanen A.C.: Riesz Spaces II. North-Holland, Amsterdam, 1983. MR 0704021 | Zbl 0519.46001
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