  The usual one third trick allows to reduce problems involving general cubes
to a countable family. Moreover, this covering lemma uses only dyadic cubes,
which allows to use nice martingale properties in harmonic analysis problems.
We consider alternatives to this technique in spaces equipped with
nonhomogeneous measures. This entails additional difficulties which forces us
to consider martingale filtrations that are not regular. The dyadic covering
that we find can be used to clarify the relationship between martingale BMO
spaces and the most natural BMO space in this setting, which is the space RBMO
introduced by Tolsa.
