The measure extension problem for vector lattices

Wright, J. D. Maitland

Annales de l'Institut Fourier, Tome 21 (1971) no. 4, pp. 65-85.

Soit $V$ un espace vectoriel ordonné $\sigma$-réticulé. Si toute prémesure à valeurs dans $V$, définie sur une algèbre de sous-ensembles de n’importe quel ensemble $X$ admet une extension $\sigma$-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 $\sigma$-distributif.

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.

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