Geodesic
Isometries of spaces of $LOG$-integrable functions
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 218-226

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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 $(\Omega_2, \mu_2)$.
Keywords: $F$-spaces, isometries, Boolean algebras, measure preserving isomorphisms, log-integrable functions. 
@article{SEMR_2020_17_a126,
     author = {R. Abdullaev and V. Chilin and B. Madaminov},
     title = {Isometries of spaces of $LOG$-integrable functions},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {218--226},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a126/}
}
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R. Abdullaev; V. Chilin; B. Madaminov. Isometries of spaces of $LOG$-integrable functions. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 218-226. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a126/