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 ) = ess sup_{x∈(0,1)} ρ_{p(x)} (δ(x)f (·)), where ρ_{p(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 EXP_α 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.