Integration of Interval Functions

An interval function Y assigns an interval Y(x) = (y(x), y(x)] in the extended real number system to each x in its interval $X = [a,b]$ of definition. The integral of Y over $[a,b]$ is taken to be the interval ∫ₐᵇ Y(x)dx = [0.3em{-}∫ ₐᵇ 0.3em{-}y (x)dx, ₐᵇ y(x)], where 0.3em{-}∫ ₐᵇ 0.3em{-}y (x) is the lower Darboux integral of the lower endpoint function 0.3em{-}y, and ∫ ₐᵇ y(x)dx is the upper Darboux integral of the upper endpoint function y. Since these Darboux integrals always exist in the extended real number system, it follows that all interval functions are integrable, no matter how nasty the endpoint functions 0.3em{-}y, y are. The interval integral defined in this way includes the interval integral of R. E. Moore as the special case that 0.3em{-}y, y are continuous, and hence Riemann integrable. In addition to a construction of the interval integral in a form suitable for numerical approximation, some of its basic properties and other implications and applications of its definition are presented. The theory of interval integration given here supplies a previously lacking mathematical foundation for the numerical solution of integral equations by interval methods.