There is a wellknown result stating that if $\mu_{n}$ is a sequence of uniformly bounded measures on a compact set $E$ of $\mathbb{R}^{m}$, then there is a subsequence $\mu_{n_{j}}$ that converges weakly to some measure $\mu$ on $E$. Now, suppose $D$ is a bounded domain of $\mathbb{R}^{m}$ with $m>1$ and $y$ a point on the boundary $\partial D$ of $D$. Take a sequence $x_{n}$ in $D$ that converges to $y$ and consider the sequence $\omega(x_{n},D)$ (harmonic measure of $D$ at $x_{n}$. This sequence has a subsequence that converges weakly to some $\mu$ on $\partial D$. My question is: is $\mu$ known? What do we know about it?

3$\begingroup$ Usually, this is the Dirac measure at $y$. (Unless $y$ is irregular for the Dirichlet problem.) $\endgroup$ – Mateusz Kwaśnicki Oct 30 '19 at 13:18

$\begingroup$ Thanks. Now suppose that for some continuous function $f$ on $\partial D$, there is a subset $A$ of $\partial D$ that does NOT contain $y$ and $$\int_{\partial D}fd\omega(x_{n},D)=\int_{A}fd\omega(x_{n},D).$$ Can $y$ be steel a regular point and the subsequence converge to $f(y)$?! $\endgroup$ – M. Rahmat Oct 30 '19 at 14:16

$\begingroup$ Sorry, I do not get your comment. Set $f = 0$ on $\partial D$ (or at least on $(\partial D) \setminus A$). Your condition is then obviously satisfied, but why would it imply anything about regularity of a boundary point $y$? $\endgroup$ – Mateusz Kwaśnicki Oct 30 '19 at 21:44

$\begingroup$ Suppose for instance that $A$ is relatively close in $\partial D$. Then the above integral of the right equals $$\int_{\partial D}\chi_{A}fd\omega(x_{n},D)\to\chi_{A}(y)f(y)=0$$ for all continuous function $f$, because $y$ doesn't belong to $A$. $\endgroup$ – M. Rahmat Oct 30 '19 at 22:32

$\begingroup$ Your last comment states the following: if $A$ is a closed subset of $\partial D$ and $y \in (\partial D) \setminus A$, then the solution of the Dirichlet problem with boundary data $\chi_A f$ converges to zero at $y$. This statement is equivalent to $y$ being regular for the Dirichlet problem in $D$. And what is the question? $\endgroup$ – Mateusz Kwaśnicki Oct 30 '19 at 22:50