An axiomatic integral and a multivariate mean value theorem

TL;DR

This paper establishes minimal axioms for an abstract integral to lie within the convex hull of the integrand and proves a multivariate mean value theorem for continuous functions over path-connected spaces.

Contribution

It introduces a set of axioms ensuring the integral's convex hull inclusion and proves a multivariate mean value theorem for continuous vector-valued functions.

Findings

Integral can be expressed as a convex combination of at most n points.

Provides minimal conditions for convex hull inclusion.

Establishes a multivariate mean value theorem.

Abstract

In order to investigate minimal sufficient conditions for an abstract integral to belong to the convex hull of the integrand, we propose a system of axioms under which it happens. If the integrand is a continuous $R^n$-valued function over a path connected topological space, we prove that any such integral can be represented as a convex combination of values of the integrand in at most $n$ points, which yields an ultimate multivariate mean value theorem.