Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 203 / 2 / d / Solution

Let ${\tau }_{\mathbb{D}}$ be the first hit of the closed unit disc. The conformal map $z\mapsto 1/z$ sends its exterior to the punctured unit disc and sends $iy$ to $-i/y$. By conformal invariance, the hitting distribution on the unit circle is harmonic measure viewed from $-i/y$. As $y\to \infty$, this point tends to zero, where harmonic measure is normalized arc length by rotational invariance. Hence for every Borel $E\subset \partial \mathbb{D}$,