Abstract
We consider the Banach function spaces, called fully measurable grand Lebesgue spaces, associated with the function norm where denotes the norm of the Lebesgue space of exponent , and and are measurable functions over a measure space , , and almost everywhere. We prove that every such space can be expressed equivalently replacing and with functions defined everywhere on the interval , 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 , the class of fully measurable Lebesgue spaces coincides with the class of generalized grand Lebesgue spaces introduced by Capone, Formica, and Giova.
Let be a finite measure space, and let be the set of all measurable functions on with values in , be the subset of the nonnegative functions, and be the subset of the real valued functions.
Let be a measure space, and define similarly as above. Let , such that and for -almost every . Following [1], for , we set whereThen is a Banach function norm, and the associated Banach function space is called fully measurable grand Lebesgue space. This definition is one of the most abstract generalizations of the original grand Lebesgue spaces originated in the paper by Iwaniec and Sbordone [2].
Given as above, let us set so that . In the following, we exclude the noninteresting case of constant , that is, the case , because in such case we know (see [1]) that .
Given a function on that is positive and decreasing (i.e., ), we consider the Banach function norm:
Theorem 1. Let , be such that is nonconstant, , and –almost everywhere. Then there exists a positive, decreasing, and left-continuous function on , such that
Some comments on this result are in order. If , coincide with the interval endowed with the Lebesgue measure, and if , then the class of spaces associated with the Banach function norm coincides with the class of spaces associated with the Banach function norm , where is increasing. In fact, substituting , we have In other words, Theorem 1 shows that the class of fully measurable Lebesgue spaces introduced in [1], where both and coincide with the interval , coincides in the case of bounded exponents with the class of spaces introduced in [3]. As a consequence, it does make no difference whether one defines the fully measurable Lebesgue spaces with measurable as above or with increasing defined on the interval and such that . This fact has a couple of consequences to be stressed.
First, to consider “essential suprema" for generalized grand Lebesgue spaces, and also for their associate spaces (which are called generalized small Lebesgue spaces), does not lead to more generality. Second, the attempts to look for the associate spaces of the fully measurable Lebesgue spaces, made in [4, 5], both solve completely the characterization of the duality. Our result may be relevant also for the weighted variant of the fully measurable Lebesgue spaces which recently attracted much interest ([6–9]).
Proof of Theorem 1. Let us set Then is positive, decreasing, and left-continuous. Let . Let , and let be such that For any , let be defined by Then is measurable and has strictly positive -measure. For any we have, by Hölder’s inequality, from which Therefore, for any , and hence On the other hand, since for -almost every , and hence
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.