  In $L^2$ extension theorems from a singular hypersurface in a complex
manifold, important roles are played by certain measures such as the Ohsawa
measure which determine when a given function can be extended. We show that the
singularity of the Ohsawa measure can be identified in terms of singularity of
pairs from algebraic geometry. Using this, we give an analytic proof of the
inversion of adjunction in this setting. Then these considerations enable us to
compare various positive and negative results on $L^2$ extension from singular
hypersurfaces. In particular, we generalize a recent negative result of Guan
and Li which places limitations on strengthening such $L^2$ extension by
employing a less singular measure in the place of the Ohsawa measure.
