Cauchy's Equation and Sufficient Statistics on Arcwise Connected Spaces

Abstract

We show that a measure-theoretic extension of Cauchy's functional equation, namely, $g(x_1) + g(x_2) = h(f(x_1, x_2))$ a.e., for real-valued functions defined on measure spaces equipped with a "reasonably compatible" arcwise connected topology is equivalent to a theorem which characterizes one-parameter exponential families on such measure spaces in terms of a real-valued sufficient statistic.