Journal of the
Korean Mathematical Society JKMS

Consider a finite measure space $(\Ome,\Sig,\mu)$ and a Banach space $X(\mu)$ consisting of (equivalence classes of) real measurable functions defined on $\Ome$ such that $f\chi_A \in X(\mu) $ and $ \|f\chi_A \| \leq \|f\|, \ \pt f \in X(\mu), \ A \in \Sig$. We prove that if it satisfies the subsequence property, then it is an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm.

Keywords: measure space, space of measurable functions, order, Banach function space

MSC numbers: 46E30, 46B42

Supported by: All the authors were supported by Ministerio de Ciencia, Innovaci´on y Universidades (Spain), Agencia Estatal de Investigaciones, and FEDER. J. M. Calabuig and M. Fern´andez-Unzueta under project MTM2014-53009-P. F. Galaz-Fontes under project MTM2009-14483-C02-01 and E. A. S´anchez P´erez under project MTM2016-77054-C2-1-P. M. Fern´andez-Unzueta was also supported by CONACYT 284110.