Let E subset of Rn+1, n >= 2, be a uniformly rectifiable set of dimension n. Then bounded harmonic functions in Omega := Rn+1 E satisfy Carleson measure estimates and are epsilon-approximable. Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute continuity of harmonic measure and surface measure.
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