We introduce a distance in the space of fully-supported probability measures on one-dimensional symbolic spaces. We compare this distance to the d-distance and we prove that in general they are not comparable. Our projective distance is inspired on Hilbert's projective metric, and in the framework of g-measures, it allows to assess the continuity of the entropy at g-measures satisfying uniqueness. It also allows to relate the speed of convergence and the regularity of sequences of locally finite g-functions, to the preservation at the limit, of certain ergodic properties for the associate g-measures.
