= Solution
Write $\mathbb H=\{z:\operatorname{Im}z>0\}$. A <compact H-hull> is a bounded, relatively closed set $A\subseteq\mathbb H$ for which $D_A=\mathbb H\setminus A$ is a <simply connected domain>. “Compact” refers to its compact <closure> in $\overline{\mathbb H}$; <boundary> points on $\mathbb R$ can be adjoined without changing the remaining <domain>.
The <Riemann mapping theorem> and normalization at infinity give a unique <mapping-out function of a compact H-hull> $g_A:D_A\to\mathbb H$ with <hydrodynamic normalization at infinity>. By the <Schwarz reflection principle> outside a large disc, it has the <Laurent series>
$$
g_A(z)=z+\frac{a}{z}+O(|z|^{-2}),\qquad a\in\mathbb R.
$$
The <half-plane capacity> is the coefficient
$$
\boxed{\operatorname{hcap}(A)=a.}
$$
The empty <compact H-hull> has $g_A(z)=z$ and <half-plane capacity> zero. Nonnegativity follows from the <Brownian representation of half-plane capacity> proved next.
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