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 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 infinitedimensional vector space that overcomes some of the known limitations valid for realvalued measures. We also discuss how the filter integral can be applied to the problem of nonArchimedean integration, and we develop the iteration theory for these integrals.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.09103
 Bibcode:
 2020arXiv200409103B
 Keywords:

 Mathematics  Logic;
 Mathematics  Functional Analysis
 EPrint:
 A complete reworking of the first version