  Directed information or its variants are utilized extensively in the
characterization of the capacity of channels with memory and feedback,
nonanticipative lossy data compression, and their generalizations to networks.
In this paper, we derive several functional and topological properties of
directed information for general abstract alphabets (complete separable metric
spaces) using the topology of weak convergence of probability measures. These
include convexity of the set of consistent distributions, which uniquely define
causally conditioned distributions, convexity and concavity of directed
information with respect to the sets of consistent distributions, weak
compactness of these sets of distributions, their joint distributions and their
marginals. Furthermore, we show lower semicontinuity of directed information,
and under certain conditions we also establish continuity of directed
information. Finally, we derive variational equalities for directed
information, including sequential versions. These may be viewed as the analogue
of the variational equalities of mutual information (utilized in Blahut-Arimoto
algorithm).
  In summary, we extend the basic functional and topological properties of
mutual information to directed information. These properties are discussed in
the context of extremum problems of directed information.
