Finite approximation of measure and integration

Abstract

Digital topology is an extreme approach to constructive spatial representation in that a classical space is replaced or represented by a finite combinatorial space. This has led to a popular research area in which theory and applications are very closely related, but the question remains as to ultimately how mathematically viable this approach is, and what the formal relationship between a space and its finite representations is. Several researchers have attempted to answer this by showing that a space can be constructed as the inverse limit of its finite representations, and this construction has been developed independently in computer science and physics. Therefore, in this formal sense, finite representations of classical spaces can be arbitrarily good. The relationship between classical and finite topology is by now fairly well understood, and in this work we extend the theory by showing that measures on the classical space are approximated by measures on finite spaces.