  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.
