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.

I suspect that $\sim$ is not transitive, though I don't see how to prove it (note that the intervals $[n_i-A,n_i+A]$ can overlap with one another). But my real question is: What is the topological closure $\approx$ (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 characterization of $\approx$ should be sufficiently concrete that one can immediately deduce (for instance) that the measure $\mu$ that assigns measure 1 to each integer $n \geq 1$ (and measure 0 to everything else) is not $\approx$-equivalent to the measure $\nu$ that assigns measure 1 to each integer $n \geq 0$ (and measure 0 to everything else).

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.

This is a discretized version of Transitive closure of balanced bounded mass transport , which I believe has not received sufficient attention.

[ADDED ON APRIL 3, MODIFIED LATER THAT DAY: I proposed a disproof of the assertion that $\sim$ is transitive, as follows: "Consider a mass distribution that puts two particles of mass 1 at each odd positive integer. Send the two particles at each odd positive integer in opposite directions to the adjacent even integers. Now we see a single particle at 0 and two particles at each even positive integer. Send the two particles at each even positive integer to the adjacent odd integers. The resulting distribution has lone particles at 0 and 1 and paired particles at 3, 5, 7, etc. The end result is that 1 unit of mass has gotten shifted from 1 to 0 while the rest of the mass distribution is unchanged. Finally, note that there is no way to achieve this redistribution with a single $\sim$-operation." The last sentence is incorrect, and as Christian Remling showed, the example is not a counterexample at all.]

Inasmuch as this thread has gotten difficult too follow, and entangles two questions (is $\sim$ transitive? if not, what is the topological closure of its transitive closure?), I request that some moderator close this thread. I have raised the first of the two questions in a new post: Transitivity of balanced mass transport in Z. If the answer turns out to be negative, then I will post a revised version of the second question.