  In this paper, we give necessary and sufficient conditions for weighted $L^2$
estimates with matrix-valued measures of well localized operators. Namely, we
seek estimates of the form: \[ \| T(\mathbf{W} f)\|_{L^2(\mathbf{V})} \le
C\|f\|_{L^2(\mathbf{W})} \] where $T$ is formally an integral operator with
additional structure, $\mathbf{W}, \mathbf{V}$ are matrix measures, and the
underlying measure space possesses a filtration. The characterization we obtain
is of Sawyer-type; in particular we show that certain natural testing
conditions obtained by studying the operator and its adjoint on indicator
functions suffice to determine boundedness. Working in both the matrix weighted
setting and the setting of measure spaces with arbitrary filtrations requires
novel modifications of a T1 proof strategy; a particular benefit of this level
of generality is that we obtain polynomial estimates on the complexity of
certain Haar shift operators.
