The measure extension problem for vector lattices
Annales de l'Institut Fourier, Tome 21 (1971) no. 4, pp. 65-85.

Soit V un espace vectoriel ordonné σ-réticulé. Si toute prémesure à valeurs dans V, définie sur une algèbre de sous-ensembles de n’importe quel ensemble X admet une extension σ-additive on dit que V a la propriété d’extension (“measure extension property”). On connaît différentes conditions sur V qui impliquent cette propriété. Mais dans cet article nous obtenons des conditions nécessaires et suffisantes. Voici la caractérisation la plus utile : V a la propriété d’extension si et seulement si toute mesure définie sur les sous-ensembles de Baire d’un espace compact, et à valeurs dans V, est régulière. Nous en tirons une caractérisation purement algébrique : V a la propriété d’extension si et seulement si V est faiblement σ-distributif.

Let V be a boundedly σ-complete vector lattice. If each V-valued premeasure on an arbitrary field of subsets of an arbitrary set can be extended to a σ-additive measure on the generated σ-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 σ-distributive.

@article{AIF_1971__21_4_65_0,
     author = {Wright, J. D. Maitland},
     title = {The measure extension problem for vector lattices},
     journal = {Annales de l'Institut Fourier},
     pages = {65--85},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {21},
     number = {4},
     year = {1971},
     doi = {10.5802/aif.393},
     mrnumber = {48 #8748},
     zbl = {0223.46012},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.393/}
}
TY  - JOUR
AU  - Wright, J. D. Maitland
TI  - The measure extension problem for vector lattices
JO  - Annales de l'Institut Fourier
PY  - 1971
SP  - 65
EP  - 85
VL  - 21
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.393/
DO  - 10.5802/aif.393
LA  - en
ID  - AIF_1971__21_4_65_0
ER  - 
%0 Journal Article
%A Wright, J. D. Maitland
%T The measure extension problem for vector lattices
%J Annales de l'Institut Fourier
%D 1971
%P 65-85
%V 21
%N 4
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/aif.393/
%R 10.5802/aif.393
%G en
%F AIF_1971__21_4_65_0
Wright, J. D. Maitland. The measure extension problem for vector lattices. Annales de l'Institut Fourier, Tome 21 (1971) no. 4, pp. 65-85. doi : 10.5802/aif.393. http://archive.numdam.org/articles/10.5802/aif.393/

[1] E. E. Floyd, "Boolean algebras with pathological order properties", Pacific J. Math., 5, 687-689 (1955). | MR | Zbl

[2] P. R. Halmos, Boolean algebras, Van Nostrand (1963). | MR | Zbl

[3] P. R. Halmos, Measure theory, Van Nostrand (1950). | MR | Zbl

[4] E. Hewitt and K. Stromberg, Real and abstract analysis, Springer (1965). | Zbl

[5] A. Horn and A. Tarski, "Measures in Boolean algebras", Trans. Amer. Math. Soc., 64, 467-497 (1948). | MR | Zbl

[6] L. V. Kantorovich, B. Z. Vulich and A. G. Pinsker, Fonctional analysis in partially ordered spaces, Gostekhizdat (1950). (Russian).

[7] E. J. Mcshane. Order-preserving maps and integration processes, Annals Math. Studies, 31 (1953). | MR | Zbl

[8] K. Matthes, "Über eine Schar von Regularitätsbedingungen", Math. Nachr., 22, 93-128 (1960). | MR | Zbl

[9] K. Matthes, "Über eine Schar von Regularitätsbedingungen II", Math. Nachr., 23, 149-159 (1961). | MR | Zbl

[10] K. Matthes, "Über die Ausdehnung von N-Homomorphismen Boolescher Algebren", Z. math. Logik u. Grundl. Math., 6, 97-105 (1960). | MR | Zbl

[11] K. Matthes, "Über die Ausdehnung von N-Homomorphismen Boolescher Algebren II", Z. Math. logik u. Grundl. Math., 7, 16-19, (1961). | MR | Zbl

[12] R. Sikorski, Boolean algebras, Springer (1962) (Second edition).

[13] R. Sikorski, "On an analogy between measures and homomorphisms", Ann. Soc. Pol. Math., 23, 1-20 (1950). | MR | Zbl

[14] M. H. Stone, Boundedness properties in function lattices, Canadian J. Math., 1 176-186 (1949). | MR | Zbl

[15] J. D. Maitland Wright, "Stone-algebra-valued measures and integrals", Proc. London Math. Soc., (3), 19, 107-122 (1969). | MR | Zbl

[16] R. V. Kadison, "A representation theory for commutative topological algebra", Memoirs Amer. Math. Soc., 7 (1951). | MR | Zbl

[17] J. L. Kelley, "Measures in Boolean algebras", Pacific J. Math. 9, 1165-1171 (1959). | MR | Zbl

Cité par Sources :