Some properties of fuzzy integrable function space \(L^ 1(\mu)\)
A measurable space (X,\(\mu)\) with fuzzy measure \(\mu\) is considered [see \textit{M. Sugeno}, Theory of fuzzy integrals and its applications, Thesis, Tokyo Institute of Technology (1974)]. Fuzzy Orlicz class \(L_{\Phi}(\mu)\) and fuzzy \(\Phi\)-mean convergence are introduced and it is proved, that for every Orlicz function \(\Phi\) the space \(L_{\Phi}(\mu)\) is equal to the space \(L^ 1(\mu)\)- defined as \(L_{\Phi}(\mu)\) whenever \(\Phi (x)=x\). It is proved that convergence in fuzzy measure in the space S(\(\mu)\) of all \(\mu\)-measurable functions coincides with fuzzy \(\Phi\)-mean convergence for any Orlicz function \(\Phi\). Also necessary and sufficient conditions for equality \(L^ 1(\mu)=S(\mu)\) are given.
scientific article; zbMATH DE number 4134819