A common generalization of infinite sum, unordered sum and integral

TL;DR

This paper introduces a unified framework that generalizes infinite sums, integrals, and other mathematical concepts by extending functions on finite sets through the Hausdorff metric.

Contribution

It provides a novel approach to unify various mathematical notions under a common generalization using Hausdorff metric extensions.

Findings

Unified framework for sums and integrals

Extension of functions on finite sets via Hausdorff metric

Potential applications in analysis and measure theory

Abstract

We present a common ground for infinite sums, unordered sums, Riemann/Lebesgue integrals, arc length and some generalized means. It is based on extending functions on finite sets using Hausdorff metric in a natural way.