Isometric $F$-spaces of $log$-integrable function
Let $(Ω_i,\mathcal A_i,μ_i) $ be a measure space with finite measure $μ_i$, and let $(L_{\log}(Ω_i, \mathcal A_i,μ_i), \|\cdot\|_{\log,μ_i})$ be a $F$-space of all $\log$-integrable functions on $(Ω_i,\mathcal A_i,μ_1), \ i =1, 2 $. It is proved that $F$-spaces $(L_{\log}(Ω_1, \mathcal A_1,μ_1), \|\cdot\|_{\log,μ_1})$ \and \ $(L_{\log}(Ω_2, \mathcal A_2,μ_2), \|\cdot\|_{\log,μ_2})$ are isometric if and only if there exists a measure preserving isomorphism from $(Ω_1, \mathcal A_1,μ_1)$ onto $(Ω_2, \mathcal A_2,μ_2)$.
math.FA↗