Cauchy Transforms of Arens Bounded Measures for a Vitushkin Amendment. I

We amend the theorem of A. G. Vitushkin [J. Functional Analysis 20 (1975) 149 - 157], who solves the problem of the rational approximation by constructive methods and a quasi-geometric notation -- the so called analytic continuous capacity --, by a new sufficient condition. The proof is functional-analytic abstract and at the same time we can almost see the effects of the condition. We start with measures -- a structure bearing level beneath the AC-capacity -- and go by Cauchy transforms directly to the level of functions. Moreover we confirm a conjecture of J. Garnett [Duke Math. J. 37(1970) 689 - 699] for functions, which are continuous on the complex numbers C including infinity and analytic off a Cantor-set.