On the weak convergence of sequences of fuzzy measures and metric of fuzzy measures (Q1568498)

A fuzzy measure is considered on the \(\sigma\)-algebra \({\mathcal A}\) of Borel subsets of a metric space (i.e., \(\mu:{\mathcal A}\to [0,\infty]\), \(\mu(\emptyset)= 0\), \(\mu\) monotone and continuous from above and from below) together with the Sugeno integral \[ \int_A f d\mu= \bigvee_{\alpha\geq 0} [\alpha\wedge \mu(A\cap \{f\geq \alpha\})]. \] A sequence \((\mu_n)\) of fuzzy measures converges to a fuzzy measure \(\mu\), if \(\lim_{n\to\infty} \int_A f d\mu_n= \int_A f d\mu\) for every nonnegative continuous function \(f: X\to R\). In the paper the weak convergence is characterized by the help of a metric defined on the set of all fuzzy measures on a separable metric space.