An axiomatic integral and a multivariate mean value theorem

2015/06/18 by Milan Merkle, Merkle, Milan
Mathematics · #26B25 #26E60 #28A30 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:26B25 #msc:26E60 #msc:28A30

paper · pdf · doi:10.48550/arxiv.1506.05551

arxiv created 2015/06/18 · arxiv updated 2015/06/19

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 Rⁿ-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.

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