On Vitali-Hahn-Saks-Nikodym type theorems

Faires, Barbara T.

Annales de l'Institut Fourier, Tome 26 (1976), p. 99-114 / Harvested from Numdam

Résumé

Une algèbre booléenne ${\cal A}$ possède la propriété (I) si étant données les suites $(a_n),(b_m)$ dans ${\cal A}$ avec $a_n\le b_m$ pour tout $n,m$, il existe un élément $b$ de ${\cal A}$ tel que $a_n\le b\le b_n$ pour tout $n$. Soit ${\cal A}$ une algèbre ayant la propriété (I). On démontre que si $(\mu _n:{\cal A}\rightarrow X)$ ($X$ un espace de Banach ) est une suite de mesures fortement additives telle que $\lim \mu _n(a)$ existe pour chaque $a\in {cal A}$, alors $\mu (a)=\lim _n\mu _n(a)$ définit une mesure fortement additive $\mu :{\cal A}\rightarrow X$ 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.

Abstract

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

Détail

Publié le : 1976-01-01

MR 56 #572 | Zbl 0309.46041

DOI : https://doi.org/10.5802/aif.633

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