Fuzzy integrals and fuzzy measures with their values in complete lattices (Q579448)

With X a non-empty set and L a complete lattice the author defines L- fuzzy integral on X and an L-fuzzy measure on X independently of each other. For each L-fuzzy integral T on X, the author defines the L-fuzzy measure \(\mu_ T\) associated to it; reciprocally, for an L-fuzzy measure \(\alpha\) on X the author defines the lower and upper L-fuzzy integral with respect to \(\alpha\), denoted respectively by \(\int f d\alpha\) and \(\int^{\vee}f d\alpha\). When L is a Brouwerian and dually Brouwerian complete lattice, then it has been shown that both lower and upper L- fuzzy integrals with respect to any L-fuzzy measure are L-fuzzy integrals; the author shows that a complete lattice L is completely distributive if and only if \(\forall X\neq \emptyset\), \(\forall \alpha \in Fm(L,X)\), \(\forall f\in L^ X\) \[ \int f d\alpha =\int^{\vee}f d\alpha, \] where \(Fm(L,X)\) is the set of all L-fuzzy measures on X; finally it has been shown that for every L-fuzzy integral T on X, \(\forall f\in L^ X\) \[ T(f)=\int f d\mu_ T\text{ and } T(f)=\int^{\vee}f d\mu_ T. \]