  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 average values so determined lie in a proper extension of the
range of the original functions. 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. These
numbers 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 valid for 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.
