  We define the two dimensional Pauli operator and identify its core for
magnetic fields that are regular Borel measures. The magnetic field is
generated by a scalar potential hence we bypass the usual $\bA\in L^2_{loc}$
condition on the vector potential which does not allow to consider such
singular fields. We extend the Aharonov-Casher theorem for magnetic fields that
are measures with finite total variation and we present a counterexample in
case of infinite total variation. One of the key technical tools is a weighted
$L^2$ estimate on a singular integral operator.
