We give a proof of the vanishing of Iwasawa mu-invariant for the cyclotomic Z_p extension of a CM field, abelian over a totally real number field. The proof follows closely Sinnott's proof for abelian number fields over the rationnals. We use Pierrette Cassou-Nogu\` es's construction of p-adic L functions for totally real number fields and a multidimensional version of Sinnott's theorem for the norm of the Gamma-transform of a measure attached to a rational function. The main point is to replace Iwasawa series attached to Cassou-Noguès's p-adic L function by a natural approximation having same norm and related to a rational function in several variables.