Integration of Interval Functions
Ole Caprani, Kaj Madsen, Louis B. Rall · SIAM Journal on Mathematical Analysis · 1981
An interval function Y assigns an interval $Y(x) = (y(x),\bar y(x)]$ 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 $\int_a^b {Y(x)dx = [\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\int } _a^b \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} (x)dx,\smallint _a^b \bar y(x)]} $, where $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\int } _a^b \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} (x)$ is the lower Darboux integral of the lower endpoint function $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} $, and $\bar \int _a^b \bar y(x)dx$ is the upper Darboux integral of the upper endpoint function $\bar y$. 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 $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} $, $\bar y$ are. The interval integral defined in this way includes the interval integral of R. E. Moore as the special case that $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{y} $, $\bar y$ 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.