no. 4
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.

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     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},
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     mrnumber = {48 #8748},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.393/}
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.393/

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