On characterizations of sup-preserving functionals

Let (E, ≦) be a vector lattice and E ₊ be the set of all nonnegative elements of E. We investigate M-functionals from E ₊ into ?₊, that is functions A: E ₊ → ?₊ such that $$ \Lambda (f \vee g) = \Lambda (f) \vee \Lambda (g),\Lambda (\alpha f) = \alpha \Lambda (f) $$ for α ≧ 0 and f, g ? E ₊. Let X be a set and Σ be an algebra of subsets of X. By an M-measure we understand the function μ: Σ → ?₊ such that μ( $ \not 0 $ ) = 0 and $$ \mu (A \cup B) = \mu (A) \vee \mu (B)forA,B \in \Sigma ). $$ The main result of the paper is a Riesz type theorem. We prove that every M-functional on C(X, ?)₊ can be expressed in terms of M-measure.