Article

Full entry | PDF   (0.2 MB)
Keywords:
topological completion; locally solid $\ell$-group; topological continuity; Fatou property; order-bound topology
Summary:
Let $(G,\tau )$ be a commutative Hausdorff locally solid lattice group. In this paper we prove the following: (1) If $(G,\tau )$ has the $A$(iii)-property, then its completion $(\widehat{G},\hat{\tau })$ is an order-complete locally solid lattice group. (2) If $G$ is order-complete and $\tau$ has the Fatou property, then the order intervals of $G$ are $\tau$-complete. (3) If $(G,\tau )$ has the Fatou property, then $G$ is order-dense in $\widehat{G}$ and $(\widehat{G},\hat{\tau })$ has the Fatou property. (4) The order-bound topology on any commutative lattice group is the finest locally solid topology on it. As an application, a version of the Nikodym boundedness theorem for set functions with values in a class of locally solid topological groups is established.
References:
[1] C. D. Aliprantis: On the completion of Hausdorff locally solid Riesz spaces. Trans. Amer. Math. Soc. 196 (1974), 105–125. DOI 10.1090/S0002-9947-1974-0350372-0 | MR 0350372 | Zbl 0258.46009
[2] C. D. Aliprantis and O. Burkinshaw: A new proof of Nakano’s theorem in locally solid Riesz spaces. Math. Zeit. 144 (1975), 25–33. DOI 10.1007/BF01214405 | MR 0385510
[3] C. D. Aliprantis and O. Burkinshaw: Nakano’s theorem revisited. Michigan Math. J. 23 (1976), 173–176. DOI 10.1307/mmj/1029001670 | MR 0454574
[4] A. Avallone and A. Valente: A decomposition theorem for submeasures. Atti. Sem. Mat. Fis. Univ. Modena XLIII (1995), 81–90. MR 1338263
[5] A. Boccuto and D. Candeloro: Uniform $s$-boundedness and convergence results for measures with values in complete $\ell$-groups. J. Math. Anal. Appl. 265 (2002), 170–194. DOI 10.1006/jmaa.2001.7715 | MR 1874264
[6] F. G. Bonales, F. J. Trigos-Arrieta and R. V. Mendoza: A characterization of Pontryagin-Van Kampen duality for locally convex spaces. Topology Appl. 121 (2002), 75–89. DOI 10.1016/S0166-8641(01)00111-0 | MR 1903684
[7] N. Bourbaki: Elements of Mathematics, General Topology, Part 1. Addison-Wesley, 1966. MR 0205210 | Zbl 0301.54001
[8] W. W. Comfort, S. Hernandez and F. J. Trigos-Arrieta: Cross sections and homeomorphism classes of Abelian groups equipped with the Bohr topology. Topology Appl. 115 (2001), 215–233. MR 1847464
[9] W. W. Comfort, S. U. Raczkowski and F. J. Trigos-Arrieta: The dual group of a dense subgroup. Czech. Math. J. 54 (129) (2004), 509–533. DOI 10.1023/B:CMAJ.0000042588.07352.99 | MR 2059270
[10] L. Drewnowski: Uniform boundedness principle for finitely additive vector measures. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. et. Phys. 21 (1973), 115–118. MR 0316670 | Zbl 0248.28007
[11] D. H. Fremlin: On the completion of locally solid vector lattices. Pacific J. Math. 43 (1972), 341–347. DOI 10.2140/pjm.1972.43.341 | MR 0318832 | Zbl 0252.46016
[12] D. H. Fremlin: Topological Riesz Spaces and Measure Theory. Cambridge University Press, England, 1974. MR 0454575 | Zbl 0273.46035
[13] J. Jakubík: On the affine completeness of lattice ordered groups. Czech. Math. J. 54 (129) (2004), 423–429. DOI 10.1023/B:CMAJ.0000042381.83544.a7 | MR 2059263
[14] G. Jameson: Ordered Linear Spaces, Lecture Notes in Mathematics No. 141, Springer-Verlag, Berlin, Germany. 1970. MR 0438077
[15] J. K. Kalton: Topologies on Riesz groups and applications to measure theory. Proc. London Math. Soc. 28 (1974), 253–273. MR 0374377 | Zbl 0276.28014
[16] A. R. Khan and K. Rowlands: A decomposition theorem for submeasures. Glasgow Math. J. 26 (1985), 67–74. MR 0776678
[17] S. U. Raczkowski: Totally bounded topological group topologies on the integers. Topology Appl. 121 (2002), 63–74. DOI 10.1016/S0166-8641(01)00110-9 | MR 1903683 | Zbl 1007.22003
[18] K. D. Schmidt: Decompositions of vector measures in Riesz spaces and Banach lattices. Proc. Edinburgh Math. Soc. 29 (1986), 23–29. MR 0829177 | Zbl 0569.28011
[19] C. Swartz: The Nikodym boundedness theorem for lattice-valued measures. Arch. Math. 53 (1989), 390–393. DOI 10.1007/BF01195219 | MR 1016003 | Zbl 0661.28003
[20] C. Swartz: An Introduction to Functional Analysis. Marcel Dekker, New York, U.S.A., 1992. MR 1156078 | Zbl 0751.46002

Partner of