  Recently it has been shown that cumulants significantly simplify the analysis
of multipartite weak measurements. Here we consider the mathematical structure
that underlies this, and find that it can be formulated in terms of what we
call the moment algebra. Apart from resulting in simpler proofs, the
flexibility of this structure allows generalizations of the original results to
a number of weak measurement scenarios, including one where the weakly
interacting pointers reach thermal equilibrium with the probed system.
