This follows the book "Stratified Lie Groups and Potential Theory for Their Sub-Laplacians" by Bonfiglioli, Lanconelli, and Uguzzoni. In chapter 6, the book introduces abstract harmonic spaces and introduces the following notion in Definition 6.2.4.
Let $\mathcal{H}$ be a harmonic sheaf on $(E, \mathcal{T})$ and let $\Omega \in \mathcal{T}$ be open. A function $u: \Omega \rightarrow (-\infty, \infty]$ is called $\mathcal H$-hyperharmonic if $u$ is (1) lower semi-continuous, and (2) for every $\mathcal{H}$-regular open set $V \subset \overline{V} \subset \Omega$ one has $$u(x) \geq \int_{\partial V} u(y)d\mu_x^V(y)$$ for all $x \in V$.
My question is: the book is quite abstract in its presentation and I can't get my head around what I'm supposed to think of. I can think of what a $\mathcal H$-regular set is and what a $\mathcal H$-harmonic measure is (for instance, by the Green's function and the representation formula or by the surface mean value theorem), but I can't think of what a $\mathcal H$-hyperharmonic function looks like for, say, the usual Laplace operator $\Delta$ on Euclidean space. (If more details on how $\mathcal H$-regular is defined or what $d\mu_x^V$ are needed then I will add this.)