  We provide a consistent regularization procedure for calculating hadron
structure functions in a chiral quark model. The structure functions are
extracted from the absorptive part of the forward Compton amplitude in the
Bjorken limit. Since this amplitude is obtained as a time-ordered correlation
function its regularization is consistently determined from the regularization
of the bosonized action. We find that the Pauli-Villars regularization scheme
is most suitable because it preserves both the anomaly structure of QCD and the
leading scaling behavior of hadron structure functions in the Bjorken limit. We
show that this procedure yields the correct pion structure function. In order
to render the sum rules of the regularized polarized nucleon structure
functions consistent with their corresponding axial charges we find it
mandatory to further specify the regularization procedure. This specification
goes beyond the double subtraction scheme commonly employed when studying
static hadron properties in this model. In particular the present approach
serves to determine the regularization prescription for structure functions
whose leading moments are not given by matrix elements of local operators. In
this regard we conclude somewhat surprisingly that in this model the Gottfried
sum rule does not undergo regularization.
