Integration with filters

Abstract

We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make sense. The most relevant scenario involves algebraic structures that extend the field of rational numbers; hence, it is possible to associate to the filter integral an upper and lower standard part, which can be interpreted as upper and lower bounds on the average value of the function that one expects to observe empirically. We discuss the main properties of the filter integral and we show that it is expressive enough to represent every real integral. As an application, we define a geometric measure on an infinite-dimensional vector space that overcomes some of the known limitations of real-valued measures. We also discuss how the filter integral can be applied to the problem of non-Archimedean integration, and we develop the iteration theory for these integrals.