  We present a higher dimensional, scale-invariant version of a classical
theorem of F. and M. Riesz. More precisely, we establish scale invariant
absolute continuity of harmonic measure with respect to surface measure, along
with higher integrability of the Poisson kernel, for a domain $\Omega\subset
\mathbb{R}^{n+1},\, n\geq 2$, with a uniformly rectifiable boundary, which
satisfies the Harnack Chain condition plus an interior (but not exterior)
corkscrew condition. In a companion paper to this one [HMU], we also establish
a converse, in which we deduce uniform rectifiability of the boundary, assuming
scale invariant $L^q$ bounds, with $q>1$, on the Poisson kernel.
