A mean value theorem for systems of integrals

More than a century ago, G. Kowalewski stated that for each n continuous functions on a compact interval [a,b], there exists an n-point quadrature rule (with respect to Lebesgue measure on [a,b]), which is exact for given functions. Here we generalize this result to continuous functions with an arbitrary positive and finite measure on an arbitrary interval. The proof relies on a new version of Carathéodory's convex hull theorem, that we also prove in the paper. As an application, we give a discrete representation of second order characteristics for a family of continuous functions of a single random variable.