Isometries of spaces of $LOG$-integrable functions

We consider the $F$-space $(L_{\log}(\Omega, \mu), \|\cdot\|_{\log})$ of $\log$-integrable functions defined on measure space $(\Omega, \mu)$ with finite measure. We prove that $(L_{\log}(\Omega_1, \mu_1), \|\cdot\|_{\log})$ and $(L_{\log}(\Omega_2, \mu_2), \|\cdot\|_{\log})$ are isometric if and only if there exists a measure preserving isomorphism from $(\Omega_1, \mu_1)$ onto $(\Omega_2, \mu_2)$.