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 version of Caratheodory's convex hull theorem for a continuous curve, that we also prove in the paper. As applications, we give a representation of the covariance for two continuous functions of a random variable, and a most general version of Gruess' inequality.