Soit un espace vectoriel ordonné -réticulé. Si toute prémesure à valeurs dans , définie sur une algèbre de sous-ensembles de n’importe quel ensemble admet une extension -additive on dit que a la propriété d’extension (“measure extension property”). On connaît différentes conditions sur qui impliquent cette propriété. Mais dans cet article nous obtenons des conditions nécessaires et suffisantes. Voici la caractérisation la plus utile : 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 , est régulière. Nous en tirons une caractérisation purement algébrique : a la propriété d’extension si et seulement si est faiblement -distributif.
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.
@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}, volume = {21}, year = {1971}, pages = {65-85}, doi = {10.5802/aif.393}, mrnumber = {48 \#8748}, zbl = {0223.46012}, language = {en}, url = {http://dml.mathdoc.fr/item/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) pp. 65-85. doi : 10.5802/aif.393. http://gdmltest.u-ga.fr/item/AIF_1971__21_4_65_0/
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