A note on noneffective weights in variable Lebesgue spaces

We study noneffective weights in the framework of variable exponent Lebesgue spaces, and we find and prove a full characterization. A result of this type was known in literature for Orlicz spaces, where surprisingly it has been shown that in some cases non trivial weights are not influent when put inside the norm. A similar phenomenon has been discovered for variable Lebesgue spaces (of course the weights are different from those ones of Orlicz spaces, but have been characterized).