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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

##### MSC:
 2.8e+11 Fuzzy measure theory 3e+72 Theory of fuzzy sets, etc.
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