  We prove that a conformal mapping defined on the unit disk belongs to a
weighted Bergman space if and only if certain integrals involving the harmonic
measure converge. With the aid of this theorem, we give a geometric
characterization of conformal mappings in Hardy or weighted Bergman spaces by
studying Euclidean areas. Applying these results, we prove several consequences
for such mappings that extend known results for Hardy spaces to weighted
Bergman spaces. Moreover, we introduce a number which is the analogue of the
Hardy number for weighted Bergman spaces. We derive various expressions for
this number and hence we establish new results for the Hardy number and the
relation between Hardy and weighted Bergman spaces.
