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Journal of Function Spaces
Volume 2017, Article ID 3129186, 3 pages
https://doi.org/10.1155/2017/3129186
Research Article

Identification of Fully Measurable Grand Lebesgue Spaces

1Dipartimento di Architettura, Università di Napoli Federico II, Via Monteoliveto 3, 80134 Napoli, Italy
2Institut für Analysis, TU Dresden, 01062 Dresden, Germany
3Istituto per le Applicazioni del Calcolo “Mauro Picone”, Sezione di Napoli, Consiglio Nazionale delle Ricerche, Via Pietro Castellino 111, 80131 Napoli, Italy

Correspondence should be addressed to Alberto Fiorenza; ti.aninu@azneroif

Received 11 May 2017; Accepted 24 August 2017; Published 2 October 2017

Academic Editor: Maria Alessandra Ragusa

Copyright © 2017 Giuseppina Anatriello et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.