  We use sphericalization to study the Dirichlet problem, Perron solutions and
boundary regularity for p-harmonic functions on unbounded sets in Ahlfors
regular metric spaces. Boundary regularity for the point at infinity is given
special attention. In particular, we allow for several "approach directions"
towards infinity and take into account the massiveness of their complements. In
2005, Llorente-Manfredi-Wu showed that the p-harmonic measure on the upper half
space $R^n_+, n \ge 2$, is not subadditive on null sets when $p \neq 2$. Using
their result and spherical inversion, we create similar bounded examples in the
unit ball $B \subset R^n$ showing that the n-harmonic measure is not
subadditive on null sets when $n \ge 3$, and neither are the p-harmonic
measures in $B$ generated by certain weights depending on $p\neq 2$ and $n \ge
2$.
