  In a previous work \cite{She} we constructed measures on symbolic spaces
which satisfy an extended multifractal formalism (in the sense that Olsen's
functions $b$ and $B$ differ and that their Legendre transforms have the
expected interpretation in terms of dimensions). These measures are composed
with a Gray code and projected onto the unit interval so to get doubling
measures. Then we were able to show that the projected measure has the same
Olsen's functions as the one it comes from and that it also fulfills the
extended multifractal formalism. Here we show that the use of a Gray code is
not necessary to get these results, although dealing with non doubling
measures.
