On Vitali-Hahn-Saks-Nikodym type theorems

Faires, Barbara T.

Annales de l'Institut Fourier, Tome 26 (1976) no. 4, pp. 99-114

A Boolean algebra $𝒜$ has the interpolation property (property (I)) if given sequences $\left(a_n\right)$, $\left(b_m\right)$ in $𝒜$ with $a_n\le b_m$ for all $n,m$, there exists an element $b$ in $𝒜$ such that $a_n\le b\le b_n$ for all $n$. Let $𝒜$ denote an algebra with the property (I). It is shown that if $({\mu }_n:𝒜\to X)$ ($X$ a Banach space) is a sequence of strongly additive measures such that ${lim}_n{\mu }_n\left(a\right)$ exists for each $a\in 𝒜$, then $\mu \left(a\right)={lim}_n{\mu }_n\left(a\right)$ defines a strongly additive map from $𝒜$ to $X$ and the ${\mu }_n^{\prime }s$ are uniformly strongly additive. The Vitali-Hahn-Saks (VHS) theorem for strongly additive $X$-valued measures defined on $𝒜$ is derived from the Nikodym boundedness theorem. A proof of the VHS theorem for group-valued measures is given.

Une algèbre booléenne $𝒜$ possède la propriété (I) si étant données les suites $\left(a_n\right),\left(b_m\right)$ dans $𝒜$ avec $a_n\le b_m$ pour tout $n,m$, il existe un élément $b$ de $𝒜$ tel que $a_n\le b\le b_n$ pour tout $n$. Soit $𝒜$ une algèbre ayant la propriété (I). On démontre que si $({\mu }_n:𝒜\to X)$ ($X$ un espace de Banach ) est une suite de mesures fortement additives telle que $lim{\mu }_n\left(a\right)$ existe pour chaque $a\in calA$, alors $\mu \left(a\right)={lim}_n{\mu }_n\left(a\right)$ définit une mesure fortement additive $\mu :𝒜\to X$ et les ${\mu }_n$ sont uniformément fortement additives. Le théorème de Vitali-Hahn-Saks (VHS) pour des mesures fortement additives dans un espace Banach est déduit du théorème de Nikodym. Une preuve du théorème (VHS) pour des mesures à valeurs dans un groupe est donnée.

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