  We study sparse domination for operators defined with respect to an atomic
filtration on a space equipped with a general measure $\mu$. In the case of
Haar shifts, $L^p$-boundedness is known to require a weak regularity condition,
which we prove to be sufficient to have a sparse domination-like theorem. Our
result allows us to characterize the class of weights where Haar shifts are
bounded. A surprising novelty is that said class depends on the complexity of
the Haar shift operator under consideration. Our results are qualitatively
sharp.
