Order convergence of vector measures on topological spaces

Surjit Singh Khurana

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 1, page 19-27
  • ISSN: 0862-7959

Abstract

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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 , the algebra generated by the zero-sets of X , and μ C b ( X ) E a positive linear map. First we give a new proof that μ extends to a unique, finitely additive measure μ E + such that ν 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 are proved, which extend some known results. Also, under certain conditions, the well-known Alexandrov’s theorem about the convergent sequences of σ -additive measures is extended to the case of order convergence.

How to cite

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Khurana, 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

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