  Given a>0, we construct a weighted Lebesgue measure on R^n for which the
family of non constant curves has p-modulus zero for p\leq 1+a but the weight
is a Muckenhoupt A_p weight for p>1+a. In particular, the p-weak gradient is
trivial for small p but non trivial for large p. This answers an open question
posed by several authors. We also give a full description of the p-weak
gradient for any locally finite Borel measure on the real line.
