We present an entropy formula of Ledrappier-Young type for invariant
measures (maybe non-SRB) of $ C^2 $ endomorphisms (maybe
non-invertible and with singularities) on a compact manifold via
their inverse limit spaces. This result may be considered as the
most general form of entropy formula for a deterministic system with
an invariant measure, and a preliminary step to Eckmann-Ruelle
conjecture. As an important application, we have proved the exact
dimensionality of ergodic measures invariant under expanding maps.