Harmonic-measure asymptotic at infinity ($\pi y{\omega }_{\mathbb{H}\setminus K}(x+iy,S)\to \mathrm{Leb}(g_K(S))$)

For a compact H-hull and a Borel subset of the intrinsic boundary of a simply connected domain, hydrodynamic normalization at infinity and the Poisson kernel for the upper half-plane give this limit as $y\to \infty$ with $x/y\to 0$. After multiplying the kernel by $\pi y$, it tends pointwise to one and is eventually uniformly bounded. Dominated convergence proves the finite-measure case and Fatou's lemma proves the infinite-measure case.