An axiomatic integral and a multivariate mean value theorem

Milan Merkle

doi:10.1186/s13660-015-0866-2

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 $\mathbf {R}^{n}$ -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.

Highlights

1 Introduction and motivation The basic integral mean value theorem states that for a function X which is continuous on the interval [a, b], there exists a point t ∈ (a, b) such that
A probability P which is defined on an algebra F of subsets of the set S is purely finitely additive if ν ≡ is the only countably additive measure with the property that ν(B) ≤ P(B) for all B ∈ F
Assuming that axioms (A )-(A )-(A ) and conditions (C )-(C ) hold, EX belongs to the convex hull of the set X(S) = {X(t) | t ∈ S} ⊂ Rn

Summary

Introduction

Let E be a functional defined on S and taking values in R such that the following axioms hold. Under the system of axioms (A )-(A )-(A ) or (A )-(A )-(A ), assuming conditions (C )-(C ), the set function P defined on F with ( ) is a finitely additive probability on (S, F ).

Results

Conclusion