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 p(f) = ess sup(x is an element of(0,1)) rho(p(x))(delta(x)f(.)), where rho(p(x)) denotes the norm of the Lebesgue space of exponent p(x) (assumed measurable and possibly infinite) and (delta 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 alpha spaces (alpha > 0). We analyze the function norm and we prove a boundedness result for the Hardy-Littlewood maximal operator, via a Hardy type inequality. (C) 2014 Elsevier Inc. All rights reserved.