Using a (weak) Gibbs measure as the reference measure, this paper introduces the concept of q-topological and q-metric pressure following the approach of Carathéodory construction. Some basic properties of q-pressure are given, including the dependence of q-pressure on the parameter q and a formula for the calculation of q-topological pressure. In particular, a relationship between the q-metric pressure and the local metric pressure is established. Finally, the correlation entropy is defined with respect to any invariant measure, and a relationship is obtained between the correlation entropy and the local metric entropy.
