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

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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].
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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     author = {Dobrakov, I. and Panchapagesan, T. V.},
     title = {A simple proof of the {Borel} extension theorem and weak compactness of operators},
     journal = {Czechoslovak Mathematical Journal},
     pages = {691--703},
     publisher = {mathdoc},
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     number = {4},
     year = {2002},
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     zbl = {1023.28005},
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     url = {http://geodesic.mathdoc.fr/item/CMJ_2002__52_4_a1/}
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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/