  We investigate the unconditional basis property of martingale differences in
weighted $L^2$ spaces in the non-homogeneous situation (i.e. when the reference
measure is not doubling).
  Specifically, we prove that finiteness of the quantity $[w]_{A_2}=\sup_I \, <
w>_I < w^{-1}>_I$, defined through averages $ <\cdot >_I$ relative to the
reference measure $\nu$, implies that each martingale transform relative to
$\nu$ is bounded in $L^2(w\, d\nu)$. Moreover, we prove the linear in
$[w]_{A_2}$ estimate of the unconditional basis constant of the Haar system.
  Even in the classical case of the standard dyadic lattice in $\mathbb{R}^n$,
where the results about unconditional basis and linear in $[w]_{A_2}$ estimates
are known, our result gives something new, because all the estimates are
independent of the dimension $n$.
  Our approach combines the technique of outer measure spaces with the Bellman
function argument.
