Identification of Fully Measurable Grand Lebesgue Spaces

We consider the Banach function spaces, called fully measurable grand Lebesgue spaces, associated with the function norm ρ(f)=ess supx∈X⁡δ(x)ρp(x)(f), where ρp(x) denotes the norm of the Lebesgue space of exponent p(x), and p(·) and δ(·) are measurable functions over a measure space (X,ν), p(x)∈[1,∞], and δ(x)∈(0,1] almost everywhere. We prove that every such space can be expressed equivalently replacing p(·) and δ(·) with functions defined everywhere on the interval (0,1), decreasing and increasing, respectively (hence the full measurability assumption in the definition does not give an effective generalization with respect to the pointwise monotone assumption and the essential supremum can be replaced with the simple supremum). In particular, we show that, in the case of bounded p(·), the class of fully measurable Lebesgue spaces coincides with the class of generalized grand Lebesgue spaces introduced by Capone, Formica, and Giova