  It is proved the existence of multivalent solutions for the Riemann-Hilbert
problem in the general settings of finitely connected domains bounded by
mutually disjoint Jordan curves, measurable coefficients and measurable
boundary data. The theorem is formulated in terms of harmonic measure and
principal asymptotic values. It is also given the corresponding reinforced
criterion for domains with rectifiable boundaries stated in terms of the
natural parameter and nontangential limits. Furthemore, it is shown that the
dimension of the spaces of these solutions is infinite.
