Then we define \(\mathbf{E}_{P}(f)\) as \(\mathbf{E}_{P}(f) = \{E_{p}(f) : p\in P \}\). That is, the set of expected values for members of \(P\). The same proviso holds of \(\mathbf{E}_{P}(f)\) as held of \(P(X)\): that is, the extent to which \(\mathbf{E}_{P}(f)\) fully represents your attitude to the value of a gamble is open to question. I will often drop the subscript “\(P\)” when no ambiguity arises from doing so. Further technical details can be found in the  formal appendix.