  Bratteli--Vershik systems have been widely studied. In the context of general
0-dimensional systems, Bratteli--Vershik systems are homeomorphisms that have
Kakutani--Rohlin refinements. Bratteli diagram has a strong power to analyze
such systems. Besides this approach, general graph covers can be used to
represent any 0-dimensional system. Indeed, all 0-dimensional systems can be
described as a certain kind of sequences of graph covers that may not be
brought about by the Kakutani--Rohlin partitions. In this paper, we follow the
context of general graph covers to analyze the relations between ergodic
measures and circuits of graph covers. First, we formalize the condition for a
sequence of graph covers to represent minimal Cantor systems. In constructing
invariant measures, we deal with general compact metrizable 0-dimensional
systems. In the context of Bratteli diagrams with finite rank, it has
previously been mentioned that all ergodic measures should be limits of some
combinations of towers of Kakutani--Rohlin refinements. We demonstrate this for
general 0-dimensional case, and develop a theorem that expresses the
coincidence of the time average and the space average for ergodic measures.
Additionally, we formulate a theorem that signifies the old relation between
uniform convergence and unique ergodicity in the context of graph circuits for
general 0-dimensional systems. Unlike previous studies, in our case of general
graph covers, there arise a possibility of linear dependence of circuits. We
give a condition for a full circuit system to be linearly independent. Previous
researches also showed that the bounded combinatorics imply unique ergodicity.
We present a lemma that enables us to consider unbounded ranks of winding
matrices.
