Isometric $F$-spaces of $log$-integrable function

Abstract

Let $(\Omega_i,\mathcal A_i,\mu_i) $ be a measure space with finite measure $\mu_i$, and let $(L_{\log}(\Omega_i, \mathcal A_i,\mu_i), \|\cdot\|_{\log,\mu_i})$ be a $F$-space of all $\log$-integrable functions on $(\Omega_i,\mathcal A_i,\mu_1), \ i =1, 2 $. It is proved that $F$-spaces $(L_{\log}(\Omega_1, \mathcal A_1,\mu_1), \|\cdot\|_{\log,\mu_1})$ \and \ $(L_{\log}(\Omega_2, \mathcal A_2,\mu_2), \|\cdot\|_{\log,\mu_2})$ are isometric if and only if there exists a measure preserving isomorphism from $(\Omega_1, \mathcal A_1,\mu_1)$ onto $(\Omega_2, \mathcal A_2,\mu_2)$.