Two-step integral with respect to fuzzy measure.

*(English)*Zbl 0948.28015In the paper the two-step Choquet integral is introduced. In general, this integral differs from the standard Choquet integral and defines a new type of a functional. The authors prove the representation theorem, showing that if a fuzzy measure \(M\) (from the definition of the integral) is additive, then the two-step integral is comonotone and therefore it can be represented by the standard Sugeno integral with respect to an appropriate fuzzy measure. The reduction of the two-step Choquet integral to the single Choquet integral in some other special situations is shown.

Moreover, the two-step integration method based on the Sugeno integral is introduced. The representation theorem, showing that if the fuzzy measure \(M\) is a maxitive measure then the two-step Sugeno integral coincides with the standard Sugeno integral with respect to an appropriate fuzzy measure, is proved.

Finally, some other possible generalizations of the two-step integration method are proposed.

Moreover, the two-step integration method based on the Sugeno integral is introduced. The representation theorem, showing that if the fuzzy measure \(M\) is a maxitive measure then the two-step Sugeno integral coincides with the standard Sugeno integral with respect to an appropriate fuzzy measure, is proved.

Finally, some other possible generalizations of the two-step integration method are proposed.

Reviewer: Anna Kolesárová (Bratislava)

##### MSC:

28E10 | Fuzzy measure theory |