Sure wins, separating probabilities and the representation of linear functionals

We discuss conditions under which a convex cone K ⊂ R Ω admits a finitely additive probability m such that sup k ∈ K m ( k ) ⩽ 0 . Based on these, we characterise those linear functionals that are representable as finitely additive expectations. A version of Riesz decomposition based on this property is obtained as well as a characterisation of positive functionals on the space of integrable functions.