  We show that the set of ergodic invariant measures of a shift space with a
safe symbol (this includes all hereditary shifts) is arcwise connected when
endowed with the $d$-bar metric. As a consequence the set of ergodic measures
of such a shift is also arcwise connected in the weak-star topology and the
entropy function over this set attains all values in the interval between zero
and the topological entropy of the shift (inclusive). The latter result is
motivated by a conjecture of A.~Katok.
