# Order convergence of vector measures on topological spaces

Mathematica Bohemica (2008)

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

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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 -

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