On pointwise bounded approximation

Let G be an open subset in the extended complex plane and let A(G) denote the algebra of all functions analytic on G and continuous on . We call a domain multi-nicely connected if there is a circular domain W and a conformal map ϕ from W onto G such that the boundary value function of ϕ is univalent almost everywhere with respect to the arclength on ∂W. Suppose that every component of G is finitely connected and none of the components of G have single point boundary components. We show that for every bounded analytic function on G to be the pointwise limit of a bounded sequence of functions in A(G), it is necessary and sufficient that each component of G is multi-nicely connected and the harmonic measures of G are mutually singular. This generalizes the corresponding result of Davie for the case when the components of G are simply connected. Supported by the Research Foundation of SWUFE