Given two atomic measures $\mu$ and $\nu$ on $\mathbb{Z}$, 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 $A>0$ such that, for all $i$, $\mu_i$ and $\nu_i$ are finite, nonzero measures supported on some interval $[n_i-A,n_i+A]$, and $\mu_i$ and $\nu_i$ have the same total mass and the same center of mass.
Is $\sim$ transitive?
In Transitive closure of balanced mass transport in Z (move to close) I conjectured (and then wrongly asserted) that the answer was "no", but a reply by Christian Remling (refuting a purported counterexample of mine) has made me less sure.
The situation that interests me most is where $\mu$ and $\nu$ are uniformly bounded, in the sense that there exists $B$ such that $\mu(n)$ and $\nu(n)$ are less than $B$ for all $n$. 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.
In the case where $\sum_{n \in \mathbb{Z}} \mu(n) = \sum_{n \in \mathbb{Z}} \nu(n) = 1$, this setup is reminiscent of the theory of martingales, so I'm tagging this question pr.probability as well as co.combinatorics. Feel free to add other tags if they seem appropriate; I'm finding this question difficult to classify.
