  This a free translation with additional explanations of {\em Processus \`a
Accroissement Independants Chapitre I: La D\'ecomposition de Paul L\'evy}, by
J.L. Bretagnolle, in {\em Ecole d'Et\'e de Probabilit\'es}, Lecture Notes in
Mathematics 307, Springer 1973. The L\'evy-Khintchine representation of
infinitely divisible distributions is obtained as a by-product.
  As this proof makes use of martingale methods, it is pedagogically more
suitable for students of financial mathematics than some other approaches. It
is hoped that the end notes will also help to make the proof more accessible to
this audience.
