Fully measurable grand Lebesgue spaces

Using variable exponents, we build a new class of rearrangement-invariant Banach function spaces, independent of the variable Lebesgue spaces, whose function norm is ρ(f)=esssupx∈(0,1)ρp(x)(δ(x)f(⋅)), where ρp(x) denotes the norm of the Lebesgue space of exponent p(x) (assumed measurable and possibly infinite) and δ is measurable, too. Such class contains some known Banach spaces of functions, among which are the classical and the grand Lebesgue spaces, and the EXPα spaces (α>0). We analyze the function norm and we prove a boundedness result for the Hardy-Littlewood maximal operator, via a Hardy type inequality.