Nonlinearity of the fuzzy integral (Q1063132)

The authors introduce the notion of a fuzzy measure \(\mu\) as a real- valued set function on a \(\sigma\)-algebra of subsets of a set X, which is isotone, continuous from above and below, and satisfies \(\mu (\emptyset)=0\) and \(\mu (X)=1\). For any real-valued, measurable functions f with values in the unit interval the fuzzy integral \(\int fd\mu\) of f is defined by sup\(\{\) min\(\{\) \(\alpha\),\(\mu\) (\(\{\) \(f\geq \alpha \})\}:\alpha \in [0,1]\}\). It is shown that the fuzzy integral conicides with the Lebesgue integral of \(\mu\) if and only if \(\mu\) is \(\{\) 0,1\(\}\)- valued. This last condition is also necessary and sufficient for a linear extension of the fuzzy integral to the set of all bounded, measurable functions.