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In this work, it is proved that the set of boundedly-compact pointed metric spaces, equipped with the Gromov-
Hausdorff topology, is a Polish space. The same is done for the Gromov-Hausdorff-Prokhorov topology. This extends
previous works which consider only length spaces or discrete metric spaces. This is a measure theoretic requirement
to study random boundedly-compact pointed (measured) metric spaces, which is the main motivation of this work. In
particular, this provides a unified framework for studying random graphs, random discrete spaces and random length
spaces. The proofs use a generalization of Strassenï¿½??s theorem, presented here, which is of independent interest.
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