Harmonic space

Definition

Let $X$ be a locally compact Hausdorff space with a countable base. To each open set $U \subseteq X$, let a linear subspace $\mathcal{H}(U)$ of the real continuous functions on $U$ be given. Its elements are called harmonic on $U$. An open set $V$ with compact closure $\overline{V} \subseteq X$ is regular if the following holds for every continuous $f \colon \partial V \to \R$. There is a unique continuous $h \colon \overline{V} \to \R$ with $h|_V \in \mathcal{H}(V)$ and $h = f$ on $\partial V$, and $h \ge 0$ whenever $f \ge 0$. The pair $(X, \mathcal{H})$ is a harmonic space if:

1. $\mathcal{H}$ is a sheaf. Restrictions of harmonic functions to smaller open sets are harmonic. A continuous function on $U = \bigcup_i U_i$ that is harmonic on each $U_i$ is harmonic on $U$.
2. The regular sets form a base of the topology of $X$.
3. For every open $U$ and every increasing sequence $(h_n)$ in $\mathcal{H}(U)$ whose limit $h$ is locally bounded on $U$, the function $h$ lies in $\mathcal{H}(U)$.