  We address $L^p(\mu)\rightarrow L^p(\lambda)$ bounds for paraproducts in the
Bloom setting. We introduce certain "sparse BMO" functions associated with
sparse collections with no infinitely increasing chains, and use these to
express sparse operators as sums of paraproducts and martingale transforms --
essentially, as Haar multipliers -- as well as to obtain an equivalence of
norms between sparse operators $\mathcal{A}_\mathcal{S}$ and compositions of
paraproducts $\Pi^*_a\Pi_b$.
