  Let $({\mathcal X}, d, \mu)$ be a metric measure space and satisfy the
so-called upper doubling condition and the geometrically doubling condition. In
this paper, the authors introduce the space ${\mathop\mathrm{RBLO}}(\mu)$ and
prove that it is a subset of the known space ${\mathop\mathrm{RBMO}}(\mu)$ in
this context. Moreover, the authors establish several useful characterizations
for the space ${\mathop\mathrm{RBLO}}(\mu)$. As an application, the authors
obtain the boundedness of the maximal Calder\'on-Zygmund operators from
$L^\infty(\mu)$ to ${\mathop\mathrm{RBLO}}(\mu)$.
