Fully measurable small Lebesgue spaces

We build a new class of Banach function spaces, whose function norm isρ(p[⋅],δ[⋅](f)=inff=∑k=1∞fk⁡∑k=1∞essinfx∈(0,1)ρp(x)(δ(x)−1fk(⋅)), where ρp(x)ρp(x) denotes the norm of the Lebesgue space of exponent p(x)p(x) (assumed measurable and possibly infinite), constant with respect to the variable of f, and δ   is measurable, too. Such class contains some known Banach spaces of functions, among which are the classical and the small Lebesgue spaces, and the Orlicz space L(log⁡L)αL(log⁡L)α, α>0α>0.Furthermore we prove the following Hölder-type inequality∫01fgdt≤ρp[⋅]),δ[⋅](f)ρ(p′[⋅],δ[⋅](g), where ρp[⋅]),δ[⋅](f)ρp[⋅]),δ[⋅](f) is the norm of fully measurable grand Lebesgue spaces introduced by Anatriello and Fiorenza in [2]. For suitable choices of p(x)p(x) and δ(x)δ(x) it reduces to the classical Hölder's inequality for the spaces EXP1/αEXP1/α and L(log⁡L)αL(log⁡L)α, α>0α>0.