Isomorphisms and isometries of $F$-spaces of $log$-integrable measurable functions

TL;DR

This paper classifies the isomorphic types of F-spaces of log-integrable functions across various measure spaces and proves that these spaces are not isometric, highlighting their structural differences.

Contribution

It provides a comprehensive isomorphic classification of log-integrable F-spaces and establishes their non-isometric nature, advancing understanding of their geometric properties.

Findings

Spaces are classified up to isomorphism.

Spaces are proven to be non-isometric.

Results apply to F-spaces constructed from different measure spaces.

Abstract

In the paper, it is given isomorphic classification of $F$-spaces of $log$-integrable measurable functions constructed using different measure spaces. At the same time, it is proved that such spaces are non-isometric.