A mean value theorem for systems of integrals

TL;DR

This paper generalizes a classical mean value theorem for systems of integrals to arbitrary measures and intervals, providing new theoretical insights and applications in covariance representation and inequalities.

Contribution

It extends Kowalewski's mean value theorem to general measures and intervals, introducing a new proof via Caratheodory's convex hull theorem and applications in probability and inequalities.

Findings

Generalized mean value theorem for systems of integrals

Representation of covariance for continuous functions of a random variable

Most general version of Gruess' inequality

Abstract

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.