  The uniformization and hyperbolization transformations formulated by Bonk,
Heinonen and Koskela in \emph{"Uniformizing Gromov Hyperbolic Spaces"},
Ast\'erisque {\bf 270} (2001), dealt with geometric properties of metric
spaces. In this paper we consider metric measure spaces and construct a
parallel transformation of measures under the uniformization and
hyperbolization procedures. We show that if a locally compact roughly starlike
Gromov hyperbolic space is equipped with a measure that is uniformly locally
doubling and supports a uniformly local $p$-Poincar\'e inequality, then the
transformed measure is globally doubling and supports a global $p$-Poincar\'e
inequality on the corresponding uniformized space. In the opposite direction,
we show that such global properties on bounded locally compact uniform spaces
yield similar uniformly local properties for the transformed measures on the
corresponding hyperbolized spaces.
  We use the above results on uniformization of measures to characterize when a
Gromov hyperbolic space, equipped with a uniformly locally doubling measure
supporting a uniformly local $p$-Poincar\'e inequality, carries nonconstant
globally defined $p$-harmonic functions with finite $p$-energy.
  We also study some geometric properties of Gromov hyperbolic and uniform
spaces. While the Cartesian product of two Gromov hyperbolic spaces need not be
Gromov hyperbolic, we construct an indirect product of such spaces that does
result in a Gromov hyperbolic space. This is done by first showing that the
Cartesian product of two bounded uniform domains is a uniform domain.
