An axiomatic integral and a multivariate mean value theorem

Abstract

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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 ......-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.