Geodesic
A simple proof of the Borel extension theorem and weak compactness of operators
Czechoslovak Mathematical Journal, Tome 52 (2002) no. 4, pp. 691-703 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $T$ be a locally compact Hausdorff space and let $C_0(T)$ be the Banach space of all complex valued continuous functions vanishing at infinity in $T$, provided with the supremum norm. Let $X$ be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of $X$-valued $\sigma $-additive Baire measures on $T$ is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map $u\: C_0(T) \rightarrow X$ when $c_0 \lnot \subset X$ are obtained. The proof of the latter result is independent of the use of powerful results such as Theorem 6 of [6] or Theorem 3 (vii) of [13].
Let $T$ be a locally compact Hausdorff space and let $C_0(T)$ be the Banach space of all complex valued continuous functions vanishing at infinity in $T$, provided with the supremum norm. Let $X$ be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of $X$-valued $\sigma $-additive Baire measures on $T$ is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map $u\: C_0(T) \rightarrow X$ when $c_0 \lnot \subset X$ are obtained. The proof of the latter result is independent of the use of powerful results such as Theorem 6 of [6] or Theorem 3 (vii) of [13].
Classification : 28B05, 28C05, 28C15, 47B07
Keywords: weakly compact operator on $C_0(T)$; representing measure; lcHs-valued $\sigma $-additive Baire (or regular Borel; or regular $\sigma $-Borel) measures 
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Dobrakov, I.; Panchapagesan, T. V. A simple proof of the Borel extension theorem and weak compactness of operators. Czechoslovak Mathematical Journal, Tome 52 (2002) no. 4, pp. 691-703. http://geodesic.mathdoc.fr/item/CMJ_2002_52_4_a1/

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