# The measure extension problem for vector lattices

• Volume: 21, Issue: 4, page 65-85
• ISSN: 0373-0956

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## Abstract

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Let $V$ be a boundedly $\sigma$-complete vector lattice. If each $V$-valued premeasure on an arbitrary field of subsets of an arbitrary set can be extended to a $\sigma$-additive measure on the generated $\sigma$-field then $V$ is said to have the measure extension property. Various sufficient conditions on $V$ which ensure that it has this property are known. But a complete characterisation of the property, that is, necessary and sufficient conditions, is obtained here. One of the most useful characterisations is: $V$has the measure extension property if, and only if, each $V$-valued Baire measure on each compact Hausdorff space is regular. This leads to an intrinsic algebraic characterisation: $V$has the measure extension property if, and only if, $V$ is weakly $\sigma$-distributive.

## How to cite

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Wright, J. D. Maitland. "The measure extension problem for vector lattices." Annales de l'institut Fourier 21.4 (1971): 65-85. <http://eudml.org/doc/74062>.

@article{Wright1971,
abstract = {Let $V$ be a boundedly $\sigma$-complete vector lattice. If each $V$-valued premeasure on an arbitrary field of subsets of an arbitrary set can be extended to a $\sigma$-additive measure on the generated $\sigma$-field then $V$ is said to have the measure extension property. Various sufficient conditions on $V$ which ensure that it has this property are known. But a complete characterisation of the property, that is, necessary and sufficient conditions, is obtained here. One of the most useful characterisations is: $V$has the measure extension property if, and only if, each $V$-valued Baire measure on each compact Hausdorff space is regular. This leads to an intrinsic algebraic characterisation: $V$has the measure extension property if, and only if, $V$ is weakly $\sigma$-distributive.},
author = {Wright, J. D. Maitland},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {4},
pages = {65-85},
publisher = {Association des Annales de l'Institut Fourier},
title = {The measure extension problem for vector lattices},
url = {http://eudml.org/doc/74062},
volume = {21},
year = {1971},
}

TY - JOUR
AU - Wright, J. D. Maitland
TI - The measure extension problem for vector lattices
JO - Annales de l'institut Fourier
PY - 1971
PB - Association des Annales de l'Institut Fourier
VL - 21
IS - 4
SP - 65
EP - 85
AB - Let $V$ be a boundedly $\sigma$-complete vector lattice. If each $V$-valued premeasure on an arbitrary field of subsets of an arbitrary set can be extended to a $\sigma$-additive measure on the generated $\sigma$-field then $V$ is said to have the measure extension property. Various sufficient conditions on $V$ which ensure that it has this property are known. But a complete characterisation of the property, that is, necessary and sufficient conditions, is obtained here. One of the most useful characterisations is: $V$has the measure extension property if, and only if, each $V$-valued Baire measure on each compact Hausdorff space is regular. This leads to an intrinsic algebraic characterisation: $V$has the measure extension property if, and only if, $V$ is weakly $\sigma$-distributive.
LA - eng
UR - http://eudml.org/doc/74062
ER -

## References

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