The measure extension problem for vector lattices

J. D. Maitland Wright

Annales de l'institut Fourier (1971)

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

Abstract

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Let be a boundedly -complete vector lattice. If each -valued premeasure on an arbitrary field of subsets of an arbitrary set can be extended to a -additive measure on the generated -field then is said to have the measure extension property. Various sufficient conditions on 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: has the measure extension property if, and only if, each -valued Baire measure on each compact Hausdorff space is regular. This leads to an intrinsic algebraic characterisation: has the measure extension property if, and only if, is weakly -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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  1. [1] E. E. FLOYD, "Boolean algebras with pathological order properties", Pacific J. Math., 5, 687-689 (1955). Zbl0065.26603MR17,450h
  2. [2] P. R. HALMOS, Boolean algebras, Van Nostrand (1963). Zbl0114.01603MR29 #4713
  3. [3] P. R. HALMOS, Measure theory, Van Nostrand (1950). Zbl0040.16802MR11,504d
  4. [4] E. HEWITT and K. STROMBERG, Real and abstract analysis, Springer (1965). Zbl0137.03202
  5. [5] A. HORN and A. TARSKI, "Measures in Boolean algebras", Trans. Amer. Math. Soc., 64, 467-497 (1948). Zbl0035.03001MR10,518h
  6. [6] L. V. KANTOROVICH, B. Z. VULICH and A. G. PINSKER, Fonctional analysis in partially ordered spaces, Gostekhizdat (1950). (Russian). 
  7. [7] E. J. McSHANE. Order-preserving maps and integration processes, Annals Math. Studies, 31 (1953). Zbl0051.29301MR15,19g
  8. [8] K. MATTHES, "Über eine Schar von Regularitätsbedingungen", Math. Nachr., 22, 93-128 (1960). Zbl0098.25601MR23 #A2342
  9. [9] K. MATTHES, "Über eine Schar von Regularitätsbedingungen II", Math. Nachr., 23, 149-159 (1961). Zbl0100.31601MR25 #3356
  10. [10] K. MATTHES, "Über die Ausdehnung von N-Homomorphismen Boolescher Algebren", Z. math. Logik u. Grundl. Math., 6, 97-105 (1960). Zbl0103.02102MR23 #A1563
  11. [11] K. MATTHES, "Über die Ausdehnung von N-Homomorphismen Boolescher Algebren II", Z. Math. logik u. Grundl. Math., 7, 16-19, (1961). Zbl0116.01504MR24 #A2547
  12. [12] R. SIKORSKI, Boolean algebras, Springer (1962) (Second edition). 
  13. [13] R. SIKORSKI, "On an analogy between measures and homomorphisms", Ann. Soc. Pol. Math., 23, 1-20 (1950). Zbl0041.17804MR12,583e
  14. [14] M. H. STONE, Boundedness properties in function lattices, Canadian J. Math., 1 176-186 (1949). Zbl0032.16901MR10,546a
  15. [15] J. D. MAITLAND WRIGHT, "Stone-algebra-valued measures and integrals", Proc. London Math. Soc., (3), 19, 107-122 (1969). Zbl0186.46504MR39 #1625
  16. [16] R. V. KADISON, "A representation theory for commutative topological algebra", Memoirs Amer. Math. Soc., 7 (1951). Zbl0042.34801MR13,360b
  17. [17] J. L. KELLEY, "Measures in Boolean algebras", Pacific J. Math. 9, 1165-1171 (1959). Zbl0087.04801MR21 #7286

Citations in EuDML Documents

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  1. Peter Volauf, Extension and regularity of -group valued measures
  2. Antonio Boccuto, Domenico Candeloro, Uniform (s)-boundedness and regularity for (l)-group-valued measures
  3. Peter Maličký, The monotone limit convergence theorem for elementary functions with values in a vector lattice
  4. Beloslav Riečan, Marta Vrábelová, On the Kurzweil integral for functions with values in ordered spaces. II.
  5. Surjit Singh Khurana, Order convergence of vector measures on topological spaces
  6. Surjit Singh Khurana, Lattice-valued Borel measures. III.
  7. Antonio Boccuto, Beloslav Riečan, On the Henstock-Kurzweil integral for Riesz-space-valued functions defined on unbounded intervals
  8. Ján Šipoš, Integration in partially ordered linear spaces
  9. Peter Volauf, On the lattice group valued submeasures
  10. Peter Volauf, On extension of maps with values in ordered spaces

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