Fundamental convergence of sequences of measurable functions on fuzzy measure space (Q1292065)

A fuzzy measure is considered as a monotone, continuous (from above and from below) extended real-valued function \(m\) defined on a \(\sigma\)-algebra such that \(m(\emptyset)=0\). It is said to be asymptotically null-additive, if \(m(A_n\cup B_m) \to 0\) \((n\to \infty,\;m\to \infty)\) whenever \(m(A_n) \to 0\) and \(m(B_m)\to 0\). It is proved that if \(m\) is asymptotically null-additive, then a sequence is fundamental in \(m\) if and only if the sequence is convergent in \(m\).