  In this note, we do the following:
  a) By using Lacey's recent technique, we give an alternative proof for
Conde-Alonso and Rey's domination theorem, which states that each positive
dyadic operator of arbitrary complexity is pointwise dominated by a positive
dyadic operator of zero complexity: \[ \sum_{S\in\mathcal{S}} \langle f
\rangle^\mu_{S^{(k)}} 1_S\lesssim (k+1) \sum_{S'\in\mathcal{S}'} \langle f
\rangle^\mu_{S'} 1_{S'}. \]
  b) By following the analogue between median and mean oscillation, we extend
Lerner's local median oscillation decomposition to arbitrary (possibly
non-doubling) measures: \[\lvert f-m(f,\hat{S_0})\rvert 1_{S_0}\lesssim
\sum_{S\in\mathcal{S}} (\omega_\lambda(f;S)+\lvert m(f,S)-m(f,\hat{S}) \rvert
)1_S.\] This can be viewed as a median oscillation decomposition adapted to the
dyadic (martingale) BMO. As an application of the decomposition, we give an
alternative proof for the dyadic (martingale) John--Nirenberg inequality, and
for Lacey's domination theorem, which states that each martingale transform is
pointwise dominated by a positive dyadic operator of complexity zero.
