An axiomatic integral and a multivariate mean value theorem

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.