  We study ergodic finite and infinite measures defined on the path space $X_B$
of a Bratteli diagram $B$ which are invariant with respect to the tail
equivalence relation on $X_B$. Our interest is focused on measures supported by
vertex and edge subdiagrams of $B$. We give several criteria when a finite
invariant measure defined on the path space of a subdiagram of $B$ extends to a
finite invariant measure on $B$. Given a finite ergodic measure on a Bratteli
diagram $B$ and a subdiagram $B'$ of $B$, we find the necessary and sufficient
conditions under which the measure of the path space $X_{B'}$ of $B'$ is
positive. For a class of Bratteli diagrams of finite rank, we determine when
they have maximal possible number of ergodic invariant measures. The case of
diagrams of rank two is completely studied. We include also an example which
explicitly illustrates the proved results.
