  For continuous maps on a compact manifold M, particularly for those that do
not preserve the Lebesgue measure m, we define the observable invariant
probability measures as a generalization of the physical measures. We prove
that any continuous map has observable measures, and characterize those that
are physical in terms of the observability. We prove that there exist physical
measures whose basins cover Lebesgue a.e, if and only if the set of all
observable measures is finite or infinite numerable. We define for any
continuous map, its generalized attractors using the set of observable
invariant measures where there is no physical measure, and prove that any
continuous map defines a decomposition of the space in up to infinitely many
generalized attractors whose basins cover Lebesgue a.e. We apply the results to
the C1 expanding maps f in the circle, proving that the set of observable
measures (even if f is not C1 plus Holder, is a subset of the equilibrium
states.

Dynamical Systems