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Integration of fuzzy-valued functions. (English) Zbl 0611.28009
In this paper a generalization of the ideas of real-valued Lebesgue measurable functions and Lebesgue-integral of real-valued functions is given; the author introduces the measurability and integrability of functions which map into the non-negative fuzzy numbers. With the said purpose the author takes the help of the concept of quasi-inverses. The set of all non-negative fuzzy numbers is denoted by H(\({\bar {\mathbb{R}}}^+)\) and a suitable partial ordering has been introduced in it; the quasi-inverses of non-negative fuzzy numbers are denoted by \(H^ q({\bar {\mathbb{R}}}^+)\) and again a suitable partial ordering is also introduced in it. The author starts with an abstract measurable space (X,\({\mathcal A})\). All subspaces of \({\bar {\mathbb{R}}}\) are thought to be equipped with the \(\sigma\)-algebra of their Borel subsets, as defined; H(\({\bar {\mathbb{R}}}^+)\) can be embedded naturally into \([0,1]^{{\bar {\mathbb{R}}}^+}\) and hence it is equipped with the trace of the product \(\sigma\)-algebra; analogously \(H^ q({\bar {\mathbb{R}}}^+)\) is considered as a subspace of \({\bar {\mathbb{R}}}^{+[0,1]}.\)
After this the measurability of \(f: X\to H({\mathbb{R}}^+)\) is defined in an obvious way, the quasi-inverse \([f]^ q: X\to H^ q({\bar {\mathbb{R}}}^+)\) of a function \(f: X\to H({\mathbb{R}}^+)\) is defined along with its measurability. In usual manner comes the integrability and the author is able to prove that the newly introduced integral is a proper extension of the Lebesgue integral; he states and proves analogues of monotone convergence theorem and dominated convergence theorem.
Reviewer: S.Saha

28E10 Fuzzy measure theory
03E72 Theory of fuzzy sets, etc.