  In this paper, we develop a theory of equipped graded graphs (or Bratteli
diagrams) and an alternative theory of projective limits of finite-dimensional
simplices. An equipment is an additional structure on the graph, namely, a
system of "cotransition" probabilities on the set of its paths. The main
problem is to describe all probability measures on the path space of a graph
with given cotransition probabilities; it goes back to the problem, posed by
E.~B.~Dynkin in the 1960s, of describing exit and entrance boundaries for
Markov chains. The most important example is the problem of describing all
central measures, to which one can reduce the problems of describing states on
AF-algebras or characters on locally finite groups. We suggest an unification
of the whole theory, an interpretation of the notions of Martin, Choquet, and
Dynkin boundaries in terms of equipped graded graphs and in terms of the theory
of projective limits of simplices. In the last section, we study the new notion
of "standardness" of projective limits of simplices and of equipped Bratteli
diagrams, as well as the notion of "lacunarization."
