  The aim of this paper is to study ultralimits of pointed metric measure
spaces (possibly unbounded and having infinite mass). We prove that ultralimits
exist under mild assumptions and are consistent with the pointed measured
Gromov-Hausdorff convergence. We also introduce a weaker variant of pointed
measured Gromov-Hausdorff convergence, for which we can prove a version of
Gromov's compactness theorem by using the ultralimit machinery. This
compactness result shows that, a posteriori, our newly introduced notion of
convergence is equivalent to the pointed measured Gromov one. Another byproduct
of our ultralimit construction is the identification of direct and inverse
limits in the category of pointed metric measure spaces.
