  In $L^2$ extension theorems from an irreducible singular hypersurface in a
complex manifold, important roles are played by certain measures such as the
Ohsawa measure, which determines when a given function can be extended. In this
paper, we show that the singularity of the Ohsawa measure can be identified in
terms of algebraic geometry. Using this, we give an analytic proof of the
inversion of adjunction in this setting. 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.
