On vector measures
Annales de l'Institut Fourier, Volume 25 (1975) no. 3-4, p. 139-161

Let $ℳ$ be the Banach space of real measures on a $\sigma$-ring $\mathbf{R}$, let ${ℳ}^{\prime }$ be its dual, let $E$ be a quasi-complete locally convex space, let ${E}^{\prime }$ be its dual, and let $\mu$ be an $E$-valued measure on $\mathbf{R}$. If is shown that for any $\theta \in {ℳ}^{\prime }$ there exists an element $\int \theta \phantom{\rule{4pt}{0ex}}d\mu$ of $E$ such that $〈{x}^{\prime }\circ \mu ,\theta 〉=〈\int \theta \phantom{\rule{4pt}{0ex}}d\mu ,{x}^{\prime }〉$ for any ${x}^{\prime }\in {E}^{\prime }$ and that the map

$\theta \to \int \theta \phantom{\rule{4pt}{0ex}}d\mu :{ℳ}^{\prime }\to E$

is order continuous. It follows that the closed convex hull of $\mu \left(\mathbf{R}\right)$ is weakly compact.

Soit $ℳ$ l’espace de Banach des mesures réelles sur une tribu $\mathbf{R}$, ${ℳ}^{\prime }$ son dual, $E$ un espace localement convexe quasi-complet, ${E}^{\prime }$ son dual et $\mu$ une mesure sur $\mathbf{R}$ à valeurs dans $E$. On démontre que pour chaque $\theta \in {ℳ}^{\prime }$ il existe un élément $\int \theta \phantom{\rule{4pt}{0ex}}d\mu \in E$ tel que $〈{x}^{\prime }\circ \mu ,\theta 〉=〈\int \theta \phantom{\rule{4pt}{0ex}}d\mu ,{x}^{\prime }〉$ pour tout ${x}^{\prime }\in {E}^{\prime }$. Si $\left({\theta }_{i}{\right)}_{i\in I}$ est une famille filtrante décroissante dans ${ℳ}^{\prime }$, dont l’infimum est 0, alors le filtre des sections de $\left\{\int {\theta }_{i}\phantom{\rule{4pt}{0ex}}d\mu \mid i\in I\right\}$ converge vers 0.

@article{AIF_1975__25_3-4_139_0,
author = {Constantinescu, Corneliu},
title = {On vector measures},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Durand},
address = {28 - Luisant},
volume = {25},
number = {3-4},
year = {1975},
pages = {139-161},
doi = {10.5802/aif.576},
zbl = {0286.46044},
mrnumber = {53 \#6301},
language = {en},
url = {http://www.numdam.org/item/AIF_1975__25_3-4_139_0}
}

Constantinescu, Corneliu. On vector measures. Annales de l'Institut Fourier, Volume 25 (1975) no. 3-4, pp. 139-161. doi : 10.5802/aif.576. http://www.numdam.org/item/AIF_1975__25_3-4_139_0/

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