Outer regularity of harmonic measure

Question

Let $\Omega$ be a bounded open set in $\mathbb{R}^{n}$ and $n\geq2$. Let $\partial\Omega$ designate the boundary of $\Omega$ and $\omega(x,\Omega$ the harmonic measure of $\Omega$ at $x\in\Omega$. I have two questions 1) is harmonic measure outer regular on Borel sets meaning for all Borel set $A$ in $\partial\Omega$ and all $\epsilon >0$, there is an open set $O$ such that $$\omega(x,\Omega)(O\setminus A)<\epsilon?$$ 2) does the above hold if $A$ is only Lebesgue measurable and not Borel measurable?

Answer 1

I think the answer to the first question is yes. The reason is that the existence and unicity of harmonic measure follows from the Riesz representation theorem. This theorem also guarantees the regularity of the measure. How ever, each open $O$ depends on $x\in\Omega$. For the second question, we need to remember that any Lebesgue measurable set is almost every where Borel. But if the set $E$ is a null set, I don't know what happens to it(?).