On certain barrelled normed spaces

Valdivia, Manuel

doi:10.5802/aif.752

Valdivia, Manuel

Annales de l'Institut Fourier, Tome 29 (1979) no. 3, pp. 39-56.

Résumé
Abstract

Soit $𝒜$ une $σ$ -algèbre dans un ensemble $X$ . Si $A$ appartient à $𝒜$ , soit $e (A)$ la fonction caractéristique de $A$ . Soit $ℓ_{0}^{\infty} (X, 𝒜$ l’espace vectoriel engendré par ${e (A) : A \in 𝒜}$ avec la topologie de la convergence uniforme. On montre que si $(E_{n})$ est une suite croissante des sous-espaces de $ℓ_{0}^{\infty} (X, 𝒜)$ dont l’union est $ℓ_{0}^{\infty} (X, 𝒜)$ il existe un entier positif $p$ , telle que $E_{p}$ est un sous-espace dense et tonnelé de $ℓ_{0}^{\infty} (X, 𝒜)$ . 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 $ℓ_{0}^{\infty} (X, 𝒜$ be the linear space generated by ${e (A) : A \in 𝒜}$ 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 $ℓ_{0}^{\infty} (X, 𝒜)$ covering it, there is a positive integer $p$ such that $E_{p}$ is a dense barrelled subspace of $ℓ_{0}^{\infty} (X, 𝒜)$ , and some new results in measure theory are deduced from this fact.

MR Zbl

DOI : 10.5802/aif.752

@article{AIF_1979__29_3_39_0,
     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},
     doi = {10.5802/aif.752},
     mrnumber = {81d:46006},
     zbl = {0379.46004},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.752/}
}

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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. http://www.numdam.org/articles/10.5802/aif.752/

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