Closed Baire sets are (sometimes) zero-sets

It is a theorem essentially due to Paul Halmos [H, 51.D] that each compact Baire set is a zero-set. Kenneth A. Ross and Karl Stromberg [RS] have shown (a bit more than the fact that) if $X$ is a completely regular Hausdorff space which is locally compact and $\sigma$-compact, then each closed Baire set in $X$ is a zero-set; the same conclusion is known to hold in case $X$ is Lindelöf and a ${G_\delta }$ in $\beta X$. In the present paper we prove the following theorem, and we show how the “closed Baire set” theorems of Ross and Stromberg emerge as corollaries: If $X$ is Baire in $\beta X$ and $A$ is a closed Baire set in $X$, then $A$ is a zero-set in $X$. Finally, we indicate how our theorem, and hence those of Ross and Stromberg, can be derived from early and forthcoming work of Frolík.