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.
J. L. Denny. "Cauchy's Equation and Sufficient Statistics on Arcwise Connected Spaces." Ann. Math. Statist. 41 (2) 401 - 411, April, 1970. https://doi.org/10.1214/aoms/1177697079