On certain barrelled normed spaces
Annales de l'Institut Fourier, Tome 29 (1979) no. 3, p. 39-56
Soit 𝒜 une σ-algèbre dans un ensemble X. Si A appartient à 𝒜, soit e(A) la fonction caractéristique de A. Soit 0 (X,𝒜 l’espace vectoriel engendré par {e(A):A𝒜} avec la topologie de la convergence uniforme. On montre que si (E n ) est une suite croissante des sous-espaces de 0 (X,𝒜) dont l’union est 0 (X,𝒜) il existe un entier positif p, telle que E p est un sous-espace dense et tonnelé de 0 (X,𝒜). Quelques nouveaux résultats dans la théorie de la mesure sont déduits de ce fait.
Let 𝒜 be a σ-algebra on a set X. If A belongs to 𝒜 let e(A) be the characteristic function of A. Let 0 (X,𝒜 be the linear space generated by {e(A):A𝒜} endowed with the topology of the uniform convergence. It is proved in this paper that if (E n ) is an increasing sequence of subspaces of 0 (X,𝒜) covering it, there is a positive integer p such that E p is a dense barrelled subspace of 0 (X,𝒜), and some new results in measure theory are deduced from this fact.
@article{AIF_1979__29_3_39_0,
     author = {Valdivia, Manuel},
     title = {On certain barrelled normed spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {29},
     number = {3},
     year = {1979},
     pages = {39-56},
     doi = {10.5802/aif.752},
     zbl = {0379.46004},
     mrnumber = {81d:46006},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1979__29_3_39_0}
}
Valdivia, Manuel. On certain barrelled normed spaces. Annales de l'Institut Fourier, Tome 29 (1979) no. 3, pp. 39-56. doi : 10.5802/aif.752. https://www.numdam.org/item/AIF_1979__29_3_39_0/

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