Let M be the maximal ideal space of the Banach algebra $H^{infty}$ of bounded analytic functions on the open unit disc $ riangle$. For a positive singular measure ${mu};on;{partial riangle},;let;{L_{+}}^1(mu)$ be the set of measures v with $0;{leq};{ u};{ll};{mu};and;{{psi}_{ u}}$ the associated singular inner functions. Let $R(mu);and;R_0(mu)$ be the union sets of ${$mid$psiv$mid$;<;1};and;{$mid${psi}_{ u}$mid$;<;0};in;M;{setminus};{ riangle},;{ u};in;{L_{+}}^1(mu)$</TEX>, respectively. It is proved that if $S(mu);=;{partial riangle}$, where $S(mu)$ is the closed support set of $mu$, then $R(mu);=;R0(mu);=;M{setminus}({ riangle};{cup};M(L^{infty}(partial riangle)))$ is generated by $H^{infty};and;overline{psi_{ u}},;{ u};{in};{L_1}^{+}(mu)$. It is proved that %d{ heta}(S(mu));=;0$ if and only if there exists as Blaschke product b with zeros ${Zn}_n$ such that $R(mu);{subset};{$mid$b$mid$;<;1};and;S(mu)$</TEX> coincides with the set of cluster points of ${Zn}_n$. While, we proved that $mu$ is a sum of finitely many point measure such that $R(mu);{subset};{$mid${psi}_{lambda}$mid$;<;1};and;S(lambda);=;S(mu)$</TEX>. Also it is studied conditions on mu for which $R(mu);=;R0(mu)$.