Die Verteilung der schlichten Funktionen in einem Funktionenraum

Abstract

We consider the set of regular functions$H = \{ f:f = z + \sum\limits_{n = 2}^\infty {nb_n z^n ,|b_n |{\mathbf{ }} \leqslant 1\} {\mathbf{ }}} on{\mathbf{ }}|z|{\mathbf{ }}< {\mathbf{ }}1$. We construct a Borel measure μ and a class of outer measures μ _h onH. With these μ and μ _h we show that: μ(H∩S)=0 and μ _h (H∩S)=0, (S is the set of normed univalent functions). From μ _h (H∩S)=0 follows—forh=t ^α—that the Hausdorff—Billingsley-dimension ofH∩S is zero.