Baire and $\sigma$-Borel characterizations of weakly compact sets in $M(T)$

Abstract:

Let $T$ be a locally compact Hausdorff space and let $M(T)$ be the Banach space of all bounded complex Radon measures on $T$. Let $\mathcal {B}_o(T)$ and $\mathcal {B}_c(T)$ be the $\sigma$-rings generated by the compact $G_\delta$ subsets and by the compact subsets of $T$, respectively. The members of $\mathcal {B}_o(T)$ are called Baire sets of $T$ and those of $\mathcal {B}_c(T)$ are called $\sigma$-Borel sets of $T$ (since they are precisely the $\sigma$-bounded Borel sets of $T$). Identifying $M(T)$ with the Banach space of all Borel regular complex measures on $T$, in this note we characterize weakly compact subsets $A$ of $M(T)$ in terms of the Baire and $\sigma$-Borel restrictions of the members of $A$. These characterizations permit us to give a generalization of a theorem of Dieudonné which is stronger and more natural than that given by Grothendieck.