We consider measures on metric spaces instead of on \(R^d\) as many techniques known from \(R^d\) work for metric spaces as well, and even the notation becomes partially simpler than that one on \(R^d\) . To this end the Borel \(\sigma \) -algebra of a metric space is introduced: It is the smallest \(\sigma \) -algebra containing all open sets. For measures on the Borel \(\sigma \) -algebra of a metric space the concepts of outer and inner regularity are introduced. The main result of this chapter is Ulam’s theorem about the tightness of a finite measure on a complete separable metric space.