Anatomy of a domain of continuous random variables I

In this paper we study the family of thin probability measures on the domain A∞A∞ of finite and infinite words over a finite alphabet A  . This structure is inspired by work of Jean Goubault-Larrecq and Daniele Varacca, who recently proposed a model of continuous random variables over bounded complete domains. Their presentation leaves out many details, and also misses some motivations. In this and a related paper we attempt to fill in some of these details, and in the process, we reveal some features of their model. Our approach to constructing the thin probability measures uses domain theory, and we show the family forms a bounded complete algebraic domain over A∞A∞. In the second paper in this series, we explore using the thin probability measures to reconstruct the bounded complete domain of continuous random variables over any bounded complete domain due originally to Goubault-Larrecq and Varacca.