  This paper describes a framework in which directed information is defined on
abstract spaces. The framework is employed to derive properties of directed
information such as convexity, concavity, lower semicontinuity, by using the
topology of weak convergence of probability measures on Polish spaces. Two
extremum problems of directed information related to capacity of channels with
memory and feedback, and non-anticipative and sequential rate distortion are
analyzed showing existence of maximizing and minimizing distributions,
respectively.
