Let M be the maximal ideal space of the Banach algebra

of bounded analytic functions on the open unit disc

. For a positive singular measure

be the set of measures v with

the associated singular inner functions. Let

be the union sets of $\{

\psiv

\;<\;1\}\;and\;\{

{\psi}_{\nu}

\;<\;0\}\;in\;M\;{\setminus}\;{\triangle},\;{\nu}\;\in\;{L_{+}}^1(\mu)$, respectively. It is proved that if

, where

is the closed support set of

, then

is generated by

. It is proved that

if and only if there exists as Blaschke product b with zeros

such that $R(\mu)\;{\subset}\;{

b

\;<\;1}\;and\;S(\mu)$ coincides with the set of cluster points of

. While, we proved that

is a sum of finitely many point measure such that $R(\mu)\;{\subset}\;\{

{\psi}_{\lambda}

\;<\;1}\;and\;S(\lambda)\;=\;S(\mu)$. Also it is studied conditions on \mu for which

.