Let \(\Omega\) be a domain in Euclidean space \(\mathbb{R}^d\), \(d\geq 3\), and \(G_\Omega(\cdot,\cdot)\) the Green function for \(\Omega\). For nonnegative superharmonic functions \(u\) on \(\Omega\), the following conditions are equivalent: {\parindent=8mm \begin{itemize}\item[(i)] the only nonnegative harmonic minorant of \(u\) is \(0\); \item[(ii)] the only nonnegative subharmonic minorant of \(u\) is \(0\); \item[(iii)] there is a Borel measure \(\mu\) on \(\Omega\) such that \(u= \int G_\Omega(\cdot, y)\,d\mu(y)\). \end{itemize}} The fine topology is the coarsest topology with respect to which all superharmonic functions are continuous. In 1972 Fuglede has shown that the fine topology counterparts of (ii) and (iii) are equivalent. In the present paper it is shown that the fine topology counterparts of (i) and (ii) are not equivalent. The precise statement is as follows: There exists a nonnegative finely superharmonic function u on a fine domain such that \(0\) is the only nonnegative finely harmonic minorant of \(u\) and \(u\) is not a fine potential. The main step of the proof is the explicit construction of a domain \(\Omega\) with an irregular boundary point \(x_0\), and a minimal positive harmonic function \(u\) on \(\Omega\) with fine limit \(+\infty\) at \(x_0\), such that \[ u(x)= c\text{ fine }\lim_{y\to x_0} G_\Omega(x, y)\;(x\in\Omega) \] does not hold for any constant \(c\).