On Vitali-Hahn-Saks-Nikodym type theorems
Annales de l'Institut Fourier, Tome 26 (1976) no. 4, pp. 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},
     pages = {99--114},
     publisher = {Imprimerie Durand},
     address = {Chartres},
     volume = {26},
     number = {4},
     year = {1976},
     doi = {10.5802/aif.633},
     mrnumber = {56 #572},
     zbl = {0309.46041},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.633/}
}
TY  - JOUR
AU  - Faires, Barbara T.
TI  - On Vitali-Hahn-Saks-Nikodym type theorems
JO  - Annales de l'Institut Fourier
PY  - 1976
SP  - 99
EP  - 114
VL  - 26
IS  - 4
PB  - Imprimerie Durand
PP  - Chartres
UR  - http://www.numdam.org/articles/10.5802/aif.633/
DO  - 10.5802/aif.633
LA  - en
ID  - AIF_1976__26_4_99_0
ER  - 
%0 Journal Article
%A Faires, Barbara T.
%T On Vitali-Hahn-Saks-Nikodym type theorems
%J Annales de l'Institut Fourier
%D 1976
%P 99-114
%V 26
%N 4
%I Imprimerie Durand
%C Chartres
%U http://www.numdam.org/articles/10.5802/aif.633/
%R 10.5802/aif.633
%G en
%F 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. http://www.numdam.org/articles/10.5802/aif.633/

[1] T. Ando, Convergent sequences of finitely additive measures, Pacific J. Math., 11 (1961), 395-404. | MR | Zbl

[2] C. Bessaga and A. Pelczynski, On bases and unconditional convergence of series in Banach spaces, Studia Math., 17 (1958), 151-164. | MR | Zbl

[3] J. K. Brooks and R. S. Jewett, On finitely additive vector measures, Proc. Nat. Acad. Sci., U.S.A., 67 (1970), 1294-1298. | MR | Zbl

[4] R. B. Darst, The Vitali-Hahn-Saks and Nikodym theorems for additive set functions, Bull. Amer. Math. Soc., 76 (1970), 1297-1298. | MR | Zbl

[5] R. B. Darst, The Vitali-Hahn-Saks and Nikodym theorems, Bull. Amer. Math. Soc., 79 (1973), 758-760. | MR | Zbl

[6] J. Diestel, Applications of weak compactness and bases to vector measures and vectoriel integration, Revue Roum. Math., 18 (1973), 211-224. | MR | Zbl

[7] J. Diestel, Grothendieck spaces and vector measures, Vector and Operator Valued Measures and Applications, Academic Press, New York, 1973, 97-108. | MR | Zbl

[8] J. Diestel and B. Faires, On vector measures, Trans. Amer. Math. Soc., 198 (1974), 253-271. | MR | Zbl

[9] J. Diestel, R. Huff and B. Faires, Convergence and boundedness of measures on non-sigma complete algebras, preprint.

[10] J. Diestel and J. Uhl, Vector measures, Notes prepared at Kent State University and the University of Illinois, 1973.

[11] L. Drewnowski, Topological rings of sets, continuous set functions, integration II, Bull. Acad. Polon. Sci., Sér. Sci. Math., Astronom. et Phys., 20 (1972), 277-286. | Zbl

[12] N. Dunford and J. Schwartz, Linear Operators, Part I, Interscience, New York, 1958. | MR | Zbl

[13] A. Grothendieck, Criteria of compactness in function spaces, Amer. J. Math., 74 (1952), 168-186. | Zbl

[14] H. Hahn, Über Folgen linearer Operationen, Monatsh. für Math. und Physik, 32 (1922), 3-88. | JFM

[15] O. M. Nikodym, Sur les familles bornées de fonctions parfaitement additives d'ensemble abstrait, Monatsh. für Math. und Physik, 40 (1933), 418-426. | JFM | Zbl

[16] O. M. Nikodym, Sur les suites convergentes de fonctions parfaitement additives d'ensemble abstrait, Monatsh, für Math. und Physik, 40 (1933), 427-432. | JFM | Zbl

[17] A. Pelczynski, Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci., Sér. Sci. Math., Astr. et Phys., 10 (1962), 641-648. | MR | Zbl

[18] C. E. Rickart, Decomposition of additive set functions, Duke Math. J., 10 (1943), 653-665. | MR | Zbl

[19] H. P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory, Studia Math., 37 (1970), 13-36. | MR | Zbl

[20] S. Saks, Addition to the note on some functionals, Trans. Amer. Math. Soc., 35 (1933), 967-974. | JFM | MR | Zbl

[21] G. Seever, Measures on F-spaces, Trans. Amer. Math. Soc., 133 (1968), 267-280. | MR | Zbl

[22] J. J. Uhl Jr., Applications of a lemma of Rosenthal to vector measures and series in Banach spaces, Preprint.

[23] G. Vitali, Sull'integrazione per serie, Rend. del Circolo Mat. di Palermo, 23 (1907), 137-155. | JFM

[24] B. Faires, On Vitali-Hahn-Saks Type Theorems, Bull. Amer. Math. Soc., 80 (1974), 670-674. | MR | Zbl

Cité par Sources :