On Vitali-Hahn-Saks-Nikodym type theorems
Annales de l'Institut Fourier, Tome 26 (1976) no. 4, p. 99-114
Une algèbre booléenne 𝒜 possède la propriété (I) si étant données les suites (a n ),(b m ) dans 𝒜 avec a n b m pour tout n,m, il existe un élément b de 𝒜 tel que a n bb n pour tout n. Soit 𝒜 une algèbre ayant la propriété (I). On démontre que si (μ n :𝒜X) (X un espace de Banach ) est une suite de mesures fortement additives telle que limμ n (a) existe pour chaque acalA, alors μ(a)=lim n μ n (a) définit une mesure fortement additive μ:𝒜X et les μ 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.
A Boolean algebra 𝒜 has the interpolation property (property (I)) if given sequences (a n ), (b m ) in 𝒜 with a n b m for all n,m, there exists an element b in 𝒜 such that a n bb n for all n. Let 𝒜 denote an algebra with the property (I). It is shown that if (μ n :𝒜X) (X a Banach space) is a sequence of strongly additive measures such that lim n μ n (a) exists for each a𝒜, then μ(a)=lim n μ n (a) defines a strongly additive map from 𝒜 to X and the μ n 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.
@article{AIF_1976__26_4_99_0,
     author = {Faires, Barbara T.},
     title = {On Vitali-Hahn-Saks-Nikodym type theorems},
     journal = {Annales de l'Institut Fourier},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {26},
     number = {4},
     year = {1976},
     pages = {99-114},
     doi = {10.5802/aif.633},
     zbl = {0309.46041},
     mrnumber = {56 \#572},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1976__26_4_99_0}
}
Faires, Barbara T. On Vitali-Hahn-Saks-Nikodym type theorems. Annales de l'Institut Fourier, Tome 26 (1976) no. 4, pp. 99-114. doi : 10.5802/aif.633. https://www.numdam.org/item/AIF_1976__26_4_99_0/

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