##### Abstract

If f is analytic in a neighborhood of ∂D = {z| |z|= 1} and if K = f(∂D), then C-K has only finitely many components; moreover, if U is a bounded simply connected region of the plane, then
∂U = U_{j=0}^{n}j
where each r_{j} is a rectifiable Jordan curve and r_{i} ∩ r_{j} is a finite set whenever i ≠ j.
Let μ be a positive regular Borel measure supported on ∂D and let m denote normalized Lebesgue measure on ∂D. If L is a compact set such that ∂L ⊂ K and R(L) is a Dirichlet algebra and if ν = μof⁻¹, then the Lebesgue decomposition of ν|_{∂V} with respect to harmonic measure for L is
ν|_{∂V} = μ_{a}of⁻¹|_{∂V} + μ_{s}of⁻¹|_{∂V}
where V = intL and μ = μ_{a} + μ_{s} is the Lebesgue decomposition of μ with respect to m.
Applying Sarason’s process, we obtain P^{∞}(ν) ≠ L^{∞}(ν) if, and only if there is a Jordan curve r contained in K such that mof⁻¹|_{Γ} << μ_{a}of⁻¹|_{Γ}. If U is a unitary operator with scalar-valued spectral measure μ then f(U) is non-reductive if and only if there is a Jordan curve r ⊂ K such that mof⁻¹|_{Γ} << μ_{a}of⁻¹|_{Γ}.
Let G be a bounded region of the plane and B(H) the algebra of bounded operators in the separable Hilbert space H. If π: H^{∞}(G)→B(H) is a norm-continuous homomorphism such that π(1) = 1 and π(z) is pure subnormal then π is weak-star, weak-star continuous. Moreover, if S is a pure subnormal contraction, the S^{*n}→0 sot.