  We consider the sum rule proposed by one of us (SLA), obtained by taking the
expectation value of an axial vector commutator in a state with one pion. The
sum rule relates the pion decay constant to integrals of pion-pion cross
sections, with one pion off the mass shell. We remark that recent data on
pion-pion scattering allow a precise evaluation of the sum rule. We also
discuss the related Adler--Weisberger sum rule (obtained by taking the
expectation value of the same commutator in a state with one nucleon),
especially in connection with the problem of extrapolation of the pion momentum
off its mass shell. We find, with current data, that both the pion-pion and
pion-nucleon sum rules are satisfied to better than six percent, and we give
detailed estimates of the experimental and extrapolation errors in the closure
discrepancies.
