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

Abstract

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 − f2|dm. 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], ⊕, ⊙) is a g-semiring with an increasing generator g.