Integral Functionals of Probability that Depend Only on Mean Values

TL;DR

This paper proves that the only continuous affine functions on the real line whose integrals depend solely on the mean are characterized, highlighting a unique property of affine functions in probability measure integration.

Contribution

It establishes that affine functions are uniquely determined by their dependence on the mean value in the context of probability measure integration.

Findings

Affine functions are the only continuous functions with mean-dependent integrals.

The result characterizes a fundamental property of probability measures and affine functions.

The proof relies on properties of compactly supported probability measures.

Abstract

It is shown that affine functions are the only continuous real valued functions on R whose integrals with respect to compactly supported probability measures depend only on the mean value of the measure.