So, \(U(E)=\{w_1\}\) and \(U(U(E))=U(\{w_1\})=\emptyset\). This means that at state \(w_1\), the Inspector is unaware of \(E\), but is not unaware that he is unaware of \(E\). More generally, Dekel et al. (1998) show that there is no nontrivial unawareness operator \(U\) satisfying the following properties: