  This paper first studies the measure theoretic pressure of measures that are
not necessarily ergodic. We define the measure theoretic pressure of an
invariant measure (not necessarily ergodic) via the Carath\'{e}odory-Pesin
structure described in \cite{Pes97}, and show that this quantity is equal to
the essential supremum of the free energy of the measures in an ergodic
decomposition. To the best of our knowledge, this formula is new even for
entropy. Meanwhile, we define the measure theoretic pressure in another way by
using separated sets, it is showed that this quantity is exactly the free
energy if the measure is ergodic. Particularly, if the dynamical system
satisfies the uniform separation condition and the ergodic measures are entropy
dense, this quantity is still equal to the the free energy even if the measure
is non-ergodic. As an application of the main result, we find that the
Hausdorff dimension of an invariant measure supported on an average conformal
repeller is given by the zero of the measure theoretic pressure of this
measure. Furthermore, if a hyperbolic diffeomorphism is average conformal and
volume-preserving, the Hausdorff dimension of any invariant measure on the
hyperbolic set is equal to the sum of the zeros of measure theoretic pressure
restricted to stable and unstable directions.
