  Given a measure $\mu$ of polynomial growth, we refine a deep result by David
and Mattila to construct an atomic martingale filtration of
$\mathrm{supp}(\mu)$ which provides the right framework for a dyadic form of
nondoubling harmonic analysis. Despite this filtration being highly irregular,
its atoms are comparable to balls in the given metric |which in turn are all
doubling| and satisfy a weaker but crucial form of regularity. Our dyadic
formulation is effective to address three basic questions:
  i) A dyadic form of Tolsa's RBMO space which contains it.
  ii) Lerner's domination and $A_2$-type bounds for nondoubling measures.
  iii) A noncommutative form of nonhomogeneous Calder\'on-Zygmund theory.
  Our martingale RBMO space preserves the crucial properties of Tolsa's
original definition and reveals its interpolation behavior with the $L_p$ scale
in the category of Banach spaces, unknown so far. On the other hand, due to
some known obstructions for Haar shifts and related concepts over nondoubling
measures, our pointwise domination theorem via sparsity naturally deviates from
its doubling analogue. In a different direction, matrix-valued harmonic
analysis over noncommutative $L_p$ spaces has recently produced profound
applications. Our analogue for nondoubling measures was expected for quite some
time. Finally, we also find a dyadic form of the Calder\'on-Zygmund
decomposition which unifies those by Tolsa and
L\'opez-S\'anchez/Martell/Parcet.
