On the Dimension of the Space of Weakly Additive Functionals

Important demanded properties of weakly additive order-preserving normalized functionals are established. Various interpretations of a weakly additive order-preserving normalized functional are given. The continuity of such a functional as a function depending on a set in a given compact space is proved. Based on these results, an example is constructed showing that the space $O(X)$ of weakly additive order-preserving normalized functionals is not embedded in any space of finite (or even countable) algebraic dimension, provided that the compact space $X$ contains more than one point.