  In this article, the historic set is divided into different level sets and we
use topological pressure to describe the size of these level sets. We give an
application of these results to dimension theory. Especially, we use
topological pressure to describe the relative multifractal spectrum of ergodic
averages and give a positive answer to the conjecture posed by L. Olsen (J.
Math. Pures Appl. {\bf 82} (2003)).
