### A Borel extension approach to weakly compact operators on ${C}_{0}\left(T\right)$

Let $X$ be a quasicomplete locally convex Hausdorff space. Let $T$ be a locally compact Hausdorff space and let ${C}_{0}\left(T\right)=\{f\phantom{\rule{0.222222em}{0ex}}T\to I$, $f$ is continuous and vanishes at infinity$\}$ be endowed with the supremum norm. Starting with the Borel extension theorem for $X$-valued $\sigma $-additive Baire measures on $T$, an alternative proof is given to obtain all the characterizations given in [13] for a continuous linear map $u\phantom{\rule{0.222222em}{0ex}}{C}_{0}\left(T\right)\to X$ to be weakly compact.