  We develop a generalized theory of (meta)equilibrium statistical mechanics in
the thermodynamic limit valid for both smooth and fractal phase spaces. In the
former case, our approach leads naturally to Boltzmann-Gibbs standard
thermostatistics while, in the latter, Tsallis thermostatistics is
straightforwardly obtained as the most appropriate formalism. We first focus on
the microcanonical ensemble stressing the importance of the limit $t \to
\infty$ on the form of the microcanonical measure. Interestingly, this approach
leads to interpret the entropic index $q$ as the box-counting dimension of the
(microcanonical) phase space when fractality is considered.
