  We solve the Dirichlet problem for the complex Monge-Amp\`ere equation on a
strictly pseudoconvex with the right hand side being a positive Borel measure
which is dominated by the Monge-Amp\`ere measure of a H\"older continuous
plurisubharmonic function. If the boundary data is continuous, then the
solution is continuous. If the boundary is H\"older continuous, then the
solution is also H\"older continuous. In particular, the answer to a question
of A. Zeriahi is always affirmative.
