  Suppose $m(\alpha)$ denotes the Mahler measure of the non-zero algebraic
number $\alpha$. For each positive real number $t$, the author studied a
version $m_t(\alpha)$ of the Mahler measure that has the triangle inequality.
The construction of $m_t$ is generic, and may be applied to a broader class of
functions defined on any Abelian group $G$. We prove analogs of known results
with an abstract function on $G$ in place of the Mahler measure. In the
process, we resolve an earlier open problem stated by the author regarding
$m_t(\alpha)$.
