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Partial orders on metric measure spaces

A partial order on the set of metric measure spaces is defined; it generalizes the Lipschitz order of Gromov. We show that our partial order is closed when metric measure spaces are equipped with the Gromov-weak topology and give a new characterization for the Lipschitz order. We will then consider some probabilistic applications. The main importance is given to the study of Fleming-Viot processes with different resampling rates. Besides that application we also consider tree-valued branching processes and two semigroups on metric measure spaces.