Construction of subharmonic function that goes to $-\infty$ on a measurable subset of the boundary In this post, one uses certain analyticity conditions of subsets of the boundary to conclude a generalized maximum principle. More precisely, they write -

Let $D \subset \Bbb C$ be a bounded domain.
Let $E$ be a subset of $\partial D$ such that there exists a subharmonic function $v<0$ on $D$ that tends to $-\infty$ when $z$ approaches $E$.
Finally let $u$ be a subharmonic function on $D$, $u \leq M$, that tends to $0$ when $z$ approaches $\partial D \setminus E$. Show that $u \leq 0$ on $D$.

In this context, my question is:

$\mathscr{(Q.1)}$ Given a measurable subset $E \subset \partial D$ does there exist a subharmonic function $v$ such that $v < 0$ on $D$ and $v$ tends to $-\infty$ while approaching $E$ ?


$\mathscr{(Q.2)}$ If the answer to $\mathscr{(Q.1)}$ is negative, then under what extra condition on $E$, we can expect the answer to be affirmative?

I was considering $E$ for some basic cases like finite set etc. But I do not seem to get control over unrestricted limit, rather only if I approach non-tangentially.
 A: A comment that got too long that shows how to do the job in the case $E$ finite and probably can be extended to a closed (or more generally $F_{\sigma}$) of measure zero on the unit circle for which Fatou example (function $f$ analytic in the unit disc with non-tangential limit $0$ precisely on $E$) works.
So assume $E$ finite and take a Jordan curve $J$ that is inside the unit disc except at the points of $E$ where it makes an acute angle say (eg you go on a parallel circle inside the unit disc and near each point of $w \in E$ you replace a small arc $\hat {z_1z_2}$ that contains $\overline {Ow} \cap \hat {z_1z_2}$ with the segments $\overline {z_1w},\overline {wz_2}$ and let $f$ a Fatou function for $E$ so analytic in the disc, bounded by $1$ and st the nontangential limit of $f$ at the points in $E$ is zero.
Then let $U$ the inside of $J$ so there is $\phi : \mathbb D \to U$ a Riemann map that extends continuously and injectively to a map $\phi : \overline {\mathbb D} \to U \cup J=\bar U$. Then clearly $g=f\circ \phi$ satisfies that $g(z) \to 0, z \to E$ unrestrictedly hence $\log |g| \to -\infty$
For sets of non zero measure, where such an approach doesn't work of course, I would look into an advanced book about subharmonic functions like Hayman's two-volume treatise as the result may be in there.
