  The space of metric measure spaces (complete separable metric spaces with a
probability measure) is becoming more and more important as state space for
stochastic processes. Of particular interest is the subspace of (continuum)
metric measure trees. Greven, Pfaffelhuber and Winter introduced the
Gromov-Prohorov metric d_{GPW} on the space of metric measure spaces and showed
that it induces the Gromov-weak topology. They also conjectured that this
topology coincides with the topology induced by Gromov's Box_1 metric. Here, we
show that this is indeed true, and the metrics are even bi-Lipschitz
equivalent. More precisely, d_{GPW}= 1/2 Box_{1/2}, and hence d_{GPW} <= Box_1
<= 2d_{GPW}. The fact that different approaches lead to equivalent metrics
underlines their importance and also that of the induced Gromov-weak topology.
  As an application, we give an easy proof of the known fact that the map
associating to a lower semi-continuous excursion the coded R-tree is Lipschitz
continuous when the excursions are endowed with the (non-separable) uniform
metric. We also introduce a new, weaker, metric topology on excursions, which
has the advantage of being separable and making the space of bounded excursions
a Lusin space. We obtain continuity also for this new topology.
