Previous |  Up |  Next


order convergence; tight and $\tau $-smooth lattice-valued vector measures; measure representation of positive linear operators; Alexandrov’s theorem
Let $X$ be a completely regular Hausdorff space, $E$ a boundedly complete vector lattice, $C_{b}(X)$ the space of all, bounded, real-valued continuous functions on $X$, $\mathcal{F}$ the algebra generated by the zero-sets of $X$, and $\mu \: C_{b}(X) \rightarrow E$ a positive linear map. First we give a new proof that $\mu $ extends to a unique, finitely additive measure $ \mu \: \mathcal{F} \rightarrow E^{+}$ such that $\nu $ is inner regular by zero-sets and outer regular by cozero sets. Then some order-convergence theorems about nets of $E^{+}$-valued finitely additive measures on $\mathcal{F}$ are proved, which extend some known results. Also, under certain conditions, the well-known Alexandrov’s theorem about the convergent sequences of $\sigma $-additive measures is extended to the case of order convergence.
[1] Aliprantis, C. D., Burkinshaw, O.: Positive Operators. Academic Press, 1985. MR 0809372
[2] Diestel, J., Uhl, J. J.: Vector Measures. Amer. Math. Soc. Surveys 15, Amer. Math. Soc., 1977. MR 0453964
[3] Kawabe, Jun: The Portmanteau theorem for Dedekind complete Riesz space-valued measures. Nonlinear Analysis and Convex Analysis, Yokohama Publ., Yokohama, 2004, pp. 149–158. MR 2144038 | Zbl 1076.28004
[4] Kawabe, Jun: Uniformity for weak order convergence of Riesz space-valued measures. Bull. Austral. Math. Soc. 71 (2005), 265–274. MR 2133410
[5] Khurana, Surjit Singh: Lattice-valued Borel measures. Rocky Mt. J. Math. 6 (1976), 377–382. MR 0399409
[6] Khurana, Surjit Singh: Lattice-valued Borel measures II. Trans. Amer. Math. Soc. 235 (1978), 205–211. MR 0460590 | Zbl 0325.28012
[7] Khurana, Surjit Singh: Topologies on spaces of continuous vector-valued functions. Trans Amer. Math. Soc. 241 (1978), 195–211. MR 0492297
[8] Khurana, Surjit Singh: A vector form of Alexanderov’s theorem. Math. Nachr. 135 (1988), 73–77. MR 0944218
[9] Lipecki, Z.: Riesz representation theorems for positive operators. Math. Nachr. 131 (1987), 351–356. MR 0908823
[10] Schaeffer, H. H.: Topological Vector Spaces. Springer, 1980.
[11] Schaeffer, H. H.: Banach Lattices and Positive Operators. Springer, 1974.
[12] Wheeler, R. F.: Survey of Baire measures and strict topologies. Expo. Math. 2 (1983), 97–190. MR 0710569 | Zbl 0522.28009
[13] Varadarajan, V. S.: Measures on topological spaces. Amer. Math. Soc. Transl. 48 (1965), 161–220.
[14] Wright, J. D. M.: Stone-algebra-valued measures and integrals. Proc. Lond. Math. Soc. 19 (1969), 107–122. MR 0240276 | Zbl 0186.46504
[15] Wright, J. D. M.: Vector lattice measures on locally compact spaces. Math. Z. 120 (1971), 193–203. MR 0293373 | Zbl 0198.47803
[16] Wright, J. D. M.: The measure extension problem for vector lattices. Ann. Inst. Fourier (Grenoble) 21 (1971), 65–85. MR 0330411 | Zbl 0215.48101
[17] Wright, J. D. M.: Measures with values in partially ordered vector spaces. Proc. Lond. Math. Soc. 25 (1972), 675–688. MR 0344413
[18] Wright, J. D. M.: An algebraic characterization of vector lattices with Borel regularity property. J. Lond. Math. Soc. 7 (1973), 277–285. MR 0333116
Partner of
EuDML logo