Full belief: \(B_i(E)=B_i^1(E)=\{w \mid p_i^w(E)=1\}\) So, full belief is defined to belief with probability one. This is a standard assumption in this literature despite a number of well-known conceptual difficulties (see Huber 2009 for an extensive discussion of this and related issues). It is sometimes useful to work with the following alternative characterization of full-belief (giving it a more “modal” flavor): Agent \(i\) believes \(E\) at state \(w\) provided all the states that \(i\) assigns positive probability to at \(w\) are in \(E\). Formally,