Perron's method is a standard construction to solve the Dirichlet Problem with continuous boundary data on domains with sufficient regularity.
My question is about Boundary data that is no longer continuous, but instead simply measurable. For the sake of simplicity, let us consider the problem on the unit disk of $\mathbb{C}$ (we can get there from any other simply connected domain through the use of a Riemann map), and the problem: $$ \begin{cases} \Delta u = 0 \text{ in } \mathbb{D} \\ u = \mathbf{1}_E \text{ on } \partial \mathbb{D} \end{cases} $$ Where here $\mathbf{1}_E$ denotes the indicator function of a Borel subset of $\partial \mathbb{D}$. Consider the Perron subfamily in this case: $$ S_{\mathbf{1}_E} = \{v \in \mathfrak{Sub}(\Omega): \limsup_{w \to z} u(w) \leq \mathbf{1}_E(z) \} $$ And the Perron solution: $$ u(z) = \sup_{v \in S_{\mathbf{1}_E}}v(z) $$ with the supremum considered pointwise.
Question: What can we say about the convergence of $u$ to the boundary data? For example, is it accurate to say $\lim_{w \to z}u(w) = \mathbf{1}_E(z)$ for almost every $z \in \partial \Omega$? If so, how would I go about proving such a statement? The saving grace seems to be that a Barrier construction can be used as is used in the standard Perron method, but I am not sure how one would flesh out the details. If anyone had an idea or a reference, either would be much appreciated.
Edit: So I think the following argument may be a step in the right direction. Let $E$ be a Borel subset of the unit circle $\partial D$. By inner, outer regularity, for all $n \in \mathbb{N}$ we have an open set $O_n$, and a compact set $K_n$ with $K_n \subset E \subset O_n$ and $\mu(O_n \setminus K_n) < \frac{1}{n}$. We may construct Urysohn functions $f_n$ with: $$ f_n(z) = \frac{d(z,O_n^c)}{d(z,O_n^c) + d(z,K_n)} $$ This function is equal to $1$ on all of $K_n$, $0$ on $\partial \Omega \setminus O_n$, and between $0,1$ on $O_n \setminus K_n$. Moreover, these functions are all continuous, and $f_n \to \mathbf{1}_E$ in $L^1$, so passing to a subsequence, $f_n \to \mathbf{1}_E$ pointwise almost everywhere. Thus, we can consider the new sequence of problems: $$ \begin{cases} \Delta u_n = 0 \text{ in } \mathbb{D} \\ u_n = f_n \text{ on } \partial \mathbb{D} \end{cases} $$ and this has a standard solution, either through Perron's method, or even simpler, the Poisson integral formula. Moroever, we can pick $f_n \to f$ monotone by replacing $f_n$ with $\max (f_1, \cdots , f_n)$ with each $f_n$ remaining continuous. Moreover, each $u_n \in S_g$, and the (pointwise) limit, $\lim_{n\to \infty} u_n(z)$ is an increasing sequence of Harmonic functions, and Harnack's inequality tells us the limit is Harmonic. This however, does not solve the issue of the boundary values. Perhaps it is here I need to use the explicit Poisson formula construction?