Weighted fully measurable grand Lebesgue spaces and the maximal theorem

Anatriello and Fiorenza (J Math Anal Appl 422:783–797, 2015) introduced the fully measurable grand Lebesgue spaces on the interval (0 , 1) ⊂ R, which contain some known Banach spaces of functions, among which there are the classical and the grand Lebesgue spaces, and the EXPα spaces (α> 0). In this paper we introduce the weighted fully measurable grand Lebesgue spaces and we prove the boundedness of the Hardy–Littlewood maximal function. Namely, let (Formula presented.), where w is a weight, 0 < δ(·) ≤ 1 ≤ p(·) < ∞, we show that if p+= ‖ p‖ ∞< + ∞, the inequality ǁMfǁp[·], δ[·], w ≤ cǁfǁp[·], δ[·], w holds with some constant c independent of f if and only if the weight w belongs to the Muckenhoupt class Ap+.