Distance function associated with the g-integral with respect to the interval-valued +-measure

Abstract

© 2017 IEEE. In the classical measure theory, the distance between two m-integrable functions f1 and f2 can be defined as L1 norm of f1 - f2, i.e. d(f1, f2) = ∫X f1 - f2dm. Instead of the Lebesgue integral, the g-integrals with respect to the interval-valued +-measure [μl, μr], where g is an increasing function, is considered, and instead of the distance f1 - f2, the function d+(f1, f2) is considered. The defined distance is an interval-valued distance function between two measurable functions which maps a nonempty set X to [a, b], where ([a, b], +, o) is a g-semiring with an increasing generator g.