Abstract
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 σ-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) → X when c 0 ⊄ 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].
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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 52, 691–703 (2002). https://doi.org/10.1023/B:CMAJ.0000027224.01146.63
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DOI: https://doi.org/10.1023/B:CMAJ.0000027224.01146.63