  We study the effects of finite spectral resolution on the magnetic field
values retrieved through the weak field approximation (WFA) from the cores of
the Mg II h & k lines. The retrieval of the line-of-sight (LOS) component of
the magnetic field, $B_{\rm LOS}$, from synthetic spectra generated in a
uniformly magnetized FAL-C atmosphere are accurate when restricted to the inner
lobes of Stokes V. As we degrade the spectral resolution, partial
redistribution (PRD) effects, that more prominently affect the outer lobes of
Stokes V, are brought into the line core through spectral smearing, degrading
the accuracy of the WFA and resulting in an inference bias, which is more
pronounced the poorer the resolution. When applied to a diverse set of spectra
emerging from a sunspot simulation, we find a good accuracy in the retrieved
$B_{\rm LOS}$ when comparing it to the model value at the height where the
optical depth in the line core is unity. The accuracy is preserved up to field
strengths of B~1500 G. Limited spectral resolution results in a small bias
toward weaker retrieved fields. The WFA for the transverse component of the
magnetic field is also evaluated. Reduced spectral resolution degrades the
accuracy of the inferences because spectral mixing results in the line
effectively probing deeper layers of the atmosphere.
