  It is well known that the space of invariant probability measures for
transitive sub-shifts of finite type is a Poulsen simplex. In this article we
prove that in the non-compact setting, for a large family of transitive
countable Markov shifts, the space of invariant sub-probability measures is a
Poulsen simplex and that its extreme points are the ergodic invariant
probability measures together with the zero measure. In particular we obtain
that the space of invariant probability measures is a Poulsen simplex minus a
vertex and the corresponding convex combinations. Our results apply to finite
entropy non-locally compact transitive countable Markov shifts and to every
locally compact transitive countable Markov shift. In order to prove these
results we introduce a topology on the space of measures that generalizes the
vague topology to a class of non-locally compact spaces, the topology of
convergence on cylinders. We also prove analogous results for suspension flows
defined over countable Markov shifts.
