= Harmonic-measure asymptotic at infinity
{title2=$\pi y\,\omega_{\mathbb H\setminus K}(x+iy,S)\to\operatorname{Leb}(g_K(S))$}
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 $x/y\to0$. 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.
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