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In this paper, several convergence results for fine $p$-(super)minimizers onquasiopen sets in metric spaces are obtained. For this purpose, we deduce aCaccioppoli-type inequality and local-to-global principles for fine$p$-(super)minimizers on quasiopen sets. A substantial part of theseconsiderations is to show that the functions belong to a suitable local fineSobolev space. We prove our results for a complete metric space equipped with adoubling measure supporting a $p$-Poincar\'e inequality with $1<p< \infty$.However, most of the results are new also for unweighted $\mathbf{R}^n$.

Convergence and local-to-global results for $p$-superminimizers on quasiopen sets