The theorem of de Finetti-Kolmogoroff-Nagumo referred to in the title of the paper states that if \(\mu \mapsto A(\mu)\), a real-valued functional defined for all probability measures \(\mu\) on a compact interval of \({\mathbb{R}}\), satisfies certain very reasonable conditions then A(\(\mu)\) is of the form \(g^{-1}\{\int g d\mu \}\) where \(g:\quad {\mathbb{R}}\to {\mathbb{R}}\) is a strictly monotonic continuous function; the authors of the present paper generalize this theorem to functionals \(\mu \mapsto A(\mu)\) where \(\mu\) varies over various classes of finitely additive, positive set functions in \({\mathbb{R}}\).