  In this paper we shall show that there exists L_0 such that for each even
integer L >= L_0 there exists $c_1 \in \rz$ for which the Julia set of $z -->
z^L + c_1$ has positive Lebesgue measure. This solves an old problem.
  Editor's note: In 1997, it was shown by Xavier Buff that there was a serious
flaw in the argument, leaving a gap in the proof. Currently (1999), the
question of polynomials with a positive measure Julia sets remains open.
