Extended multifractal formalism of some non-doubling measures

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.