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All the notions of convergence of measures that I know of are either in the purely measure-theoretic category (e.g. strong convergence, total variation), or in the topological category (e.g. weak convergence), or at most the category of metric spaces (e.g. Wasserstein distances).

Are any well-established and well-understood notions of convergence of measures specifically designed for measures on spaces that are assumed to have some differentiable structure?

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(Linear) spaces of measures are in natural duality with spaces of continuous functions; see e.g. the Riesz--Markov--Kakutani_representation theorem. Spaces of differentiable functions are in natural duality with spaces of Sobolev--Schwartz-like generalized functions/distributions, of which spaces of measures are proper subspaces. Therefore, the answer to your question is "probably no".

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  • $\begingroup$ Thank you for this. Yes, I see how this reasoning might indicate that trying to do something like a differentiable analogue of the topological notion of weak convergence of probability measures won't work (i.e. it will just end up the same as weak convergence of probability measures). But is there good reason to expect that in general, every reasonable notion of convergence of probability measures fits more-or-less into the kind of framework you've described? $\endgroup$
    – Julian Newman
    Commented Dec 19, 2019 at 22:43

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