It is well-known that one can extend the definition of Gromov-Hausdorff convergence to non-compact metric spaces by instead considering pointed Gromov-Hausdorff convergence.
There is also an equivalent notion of measured Gromov-Hausdorff convergence for compact normalized metric spaces. Can one extend this notion to pointed measured Gromov-Hausdorff convergence of non-compact metric spaces in the analogous way by working with balls in that metric space?
I'm not sure how it works with a non-compact metric measure space, can one still normalize the measure if the space is not compact?
Edit: I've looked into it and there is indeed such a notion, see work of Lott-Villani and Gigli-Mondino-Savaré if anyone is interested.