The purpose of this paper is essentially didactic: to call attention to and make explicit certain properties of measures on spaces containing a denumerable number of points with a view to justifying the conventional treatment of such spaces in probability texts. The authors show that an arbitrary \(\sigma\)-ring of subsets of a countable space is generated by its atoms and, as a consequence, that any measure on such a \(\sigma\)-ring can be extended to the power set of the space. The implications of the results for probability theory are explored. Connections with theorems of Boolean algebra are exhibited, but the proofs are in a measure-theoretic setting and do not require any knowledge of Boolean algebra.