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For 
$p\,>\,0$ and for a given set 
$E$ of type 
${{G}_{\delta }}$ in the boundary of the unit disc 
$\partial \mathbb{D}$ we construct a holomorphic function 
$f\,\in \,\mathbb{O}\left( \mathbb{D} \right)$ such that
$${{\int_{\mathbb{D}\backslash \left[ 0,\,1 \right]E}{\left| f \right|}}^{p}}\,d{{\mathfrak{L}}^{2}}\,<\,\infty \,\text{and}\,E\,=\,{{E}^{p}}\left( f \right)\,=\,\{\,z\,\in \,\partial \mathbb{D}\,:\,\int _{0}^{1}\,{{\left| f\left( tz \right) \right|}^{p}}\,dt\,=\,\infty \}.$$
In particular if a set $E$ has a measure equal to zero, then a function 
$f$ is constructed as integrable with power 
$p$ on the unit disc 
$\mathbb{D}$.
Assume that 
$\Omega$ is a Hartogs domain in 
${{\mathbb{C}}^{1+n}}$, defined as 
$\Omega =\left\{ \left( z,w \right)\,\in \,{{\mathbb{C}}^{1+n}}\,:\left| z \right|\,<\,\mu \left( w \right),w\,\in H \right\}$, where 
$H$ is an open set in 
${{\mathbb{C}}^{n}}$ and 
$\mu$ is a continuous function with positive values in $H$ such that –ln $\mu$ is a strongly plurisubharmonic function in $H$. Let 
${{\Omega }_{w}}=\Omega \cap \left( \mathbb{C}\times \left\{ w \right\} \right)$. For a given set 
$E$ contained in $H$ of the type 
${{G}_{\delta }}$ we construct a holomorphic function 
$f\in \mathbb{O}\left( \Omega \right)$ such that
$$E=\left\{ w\in {{\mathbb{C}}^{n}}:\int\limits_{{{\Omega }_{w}}}{{{\left| f\left( \cdot ,w \right) \right|}^{2}}d{{\mathfrak{L}}^{2}}=\infty } \right\}.$$
