Order convergence of vector measures on topological spaces
Mathematica Bohemica (2008)
- Volume: 133, Issue: 1, page 19-27
- ISSN: 0862-7959
Access Full Article
topAbstract
topHow to cite
topKhurana, Surjit Singh. "Order convergence of vector measures on topological spaces." Mathematica Bohemica 133.1 (2008): 19-27. <http://eudml.org/doc/250512>.
@article{Khurana2008,
abstract = {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.},
author = {Khurana, Surjit Singh},
journal = {Mathematica Bohemica},
keywords = {order convergence; tight and $\tau $-smooth lattice-valued vector measures; measure representation of positive linear operators; Alexandrov’s theorem; order convergence; tight and -smooth lattice-valued vector measures; measure representation of positive linear operators; Alexandrov's theorem},
language = {eng},
number = {1},
pages = {19-27},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Order convergence of vector measures on topological spaces},
url = {http://eudml.org/doc/250512},
volume = {133},
year = {2008},
}
TY - JOUR
AU - Khurana, Surjit Singh
TI - Order convergence of vector measures on topological spaces
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 1
SP - 19
EP - 27
AB - 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.
LA - eng
KW - order convergence; tight and $\tau $-smooth lattice-valued vector measures; measure representation of positive linear operators; Alexandrov’s theorem; order convergence; tight and -smooth lattice-valued vector measures; measure representation of positive linear operators; Alexandrov's theorem
UR - http://eudml.org/doc/250512
ER -
References
top- Positive Operators, Academic Press, 1985. (1985) MR0809372
- Vector Measures, Amer. Math. Soc. Surveys 15, Amer. Math. Soc., 1977. (1977) MR0453964
- The Portmanteau theorem for Dedekind complete Riesz space-valued measures, Nonlinear Analysis and Convex Analysis, Yokohama Publ., Yokohama, 2004, pp. 149–158. (2004) Zbl1076.28004MR2144038
- 10.1017/S0004972700038235, Bull. Austral. Math. Soc. 71 (2005), 265–274. (2005) MR2133410DOI10.1017/S0004972700038235
- 10.1216/RMJ-1976-6-2-377, Rocky Mt. J. Math. 6 (1976), 377–382. (1976) MR0399409DOI10.1216/RMJ-1976-6-2-377
- 10.1090/S0002-9947-1978-0460590-2, Trans. Amer. Math. Soc. 235 (1978), 205–211. (1978) Zbl0325.28012MR0460590DOI10.1090/S0002-9947-1978-0460590-2
- 10.1090/S0002-9947-1978-0492297-X, Trans Amer. Math. Soc. 241 (1978), 195–211. (1978) MR0492297DOI10.1090/S0002-9947-1978-0492297-X
- 10.1002/mana.19881350107, Math. Nachr. 135 (1988), 73–77. (1988) MR0944218DOI10.1002/mana.19881350107
- 10.1002/mana.19871310130, Math. Nachr. 131 (1987), 351–356. (1987) MR0908823DOI10.1002/mana.19871310130
- Topological Vector Spaces, Springer, 1980. (1980)
- Banach Lattices and Positive Operators, Springer, 1974. (1974)
- Survey of Baire measures and strict topologies, Expo. Math. 2 (1983), 97–190. (1983) Zbl0522.28009MR0710569
- 10.1090/trans2/048/10, Amer. Math. Soc. Transl. 48 (1965), 161–220. (1965) DOI10.1090/trans2/048/10
- 10.1112/plms/s3-19.1.107, Proc. Lond. Math. Soc. 19 (1969), 107–122. (1969) Zbl0186.46504MR0240276DOI10.1112/plms/s3-19.1.107
- 10.1007/BF01117493, Math. Z. 120 (1971), 193–203. (1971) Zbl0198.47803MR0293373DOI10.1007/BF01117493
- 10.5802/aif.393, Ann. Inst. Fourier (Grenoble) 21 (1971), 65–85. (1971) Zbl0215.48101MR0330411DOI10.5802/aif.393
- 10.1112/plms/s3-25.4.675, Proc. Lond. Math. Soc. 25 (1972), 675–688. (1972) MR0344413DOI10.1112/plms/s3-25.4.675
- 10.1112/jlms/s2-7.2.277, J. Lond. Math. Soc. 7 (1973), 277–285. (1973) MR0333116DOI10.1112/jlms/s2-7.2.277
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.