  With no criteria of the index type, it is proved the existence of a solution
for the Riemann-Hilbert problem in the fairly general setting of arbitrary
Jordan domains, 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 arbitrary rectifiable boundaries stated in terms of the natural parameter
and nontangential limits. Moreover, it is shown that the dimension of the
spaces of solutions is infinite.
