no. 3
On certain barrelled normed spaces
Annales de l'Institut Fourier, Volume 29 (1979) no. 3, pp. 39-56.

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 ℓ 0 ∞ (X,𝒜 be the linear space generated by {e(A):A∈𝒜} 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 ℓ 0 ∞ (X,𝒜) covering it, there is a positive integer p such that E p is a dense barrelled subspace of ℓ 0 ∞ (X,𝒜), and some new results in measure theory are deduced from this fact.

Soit 𝒜 une σ-algèbre dans un ensemble X. Si A appartient à 𝒜, soit e(A) la fonction caractéristique de A. Soit ℓ 0 ∞ (X,𝒜 l’espace vectoriel engendré par {e(A):A∈𝒜} avec la topologie de la convergence uniforme. On montre que si (E n ) est une suite croissante des sous-espaces de ℓ 0 ∞ (X,𝒜) dont l’union est ℓ 0 ∞ (X,𝒜) il existe un entier positif p, telle que E p est un sous-espace dense et tonnelé de ℓ 0 ∞ (X,𝒜). Quelques nouveaux résultats dans la théorie de la mesure sont déduits de ce fait.

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     author = {Valdivia, Manuel},
     title = {On certain barrelled normed spaces},
     journal = {Annales de l'Institut Fourier},
     pages = {39--56},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {29},
     number = {3},
     year = {1979},
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Valdivia, Manuel. On certain barrelled normed spaces. Annales de l'Institut Fourier, Volume 29 (1979) no. 3, pp. 39-56. doi : 10.5802/aif.752. https://aif.centre-mersenne.org/articles/10.5802/aif.752/

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