Given two $\sigma$-finite measures $\mu$ and $\nu$ on $\mathbb{R}^n$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ along with some constant $R>0$ such that, for all $i$, $\mu_i$ and $\nu_i$ are finite, nonzero measures supported on some ball $B_i$ of radius $R$, and $\mu_i$ and $\nu_i$ have the same total mass and the same center of mass.
The relation $\sim$ is not transitive (for a discussion of this see the companion post Transitive closure of balanced mass transport in Z (move to close)). But my real question is: What is the topological closure (relative to the total variation metric topology) of the transitive closure of $\sim$? Which is to say: How can one concretely characterize the resulting topologically closed, transitive relation?
The situation that interests me most is where $\mu$ and $\nu$ are uniformly bounded, in the sense that for every $R$ there exists $M$ such that every ball of radius $R$ has measure at most $M$. But I'm not sure how such an assumption would affect my central question, so I omitted it as an hypothesis. Feel free to assume it if the assumption gives you some traction.
If my original question seems too hard to get a handle on, one might get somewhere by initially restricting attention to $\sigma$-finite measures supported on $\mathbb{Z}^n$. The case $n=1$ is already challenging; I've raised it as a separate thread at Transitive closure of balanced mass transport in Z (move to close) .
(Feel free to add or delete tags if mine seem inappropriate; I'm finding this question difficult to classify.)
