From these concepts we can define some “summary functions” that are often used in discussions of imprecise probabilities. Often, it is assumed that your degree of belief in a proposition, \(X\), is represented by \(P(X) = \{p(X) : p\in P \}\). I will adopt this notational convention, with the proviso that I don’t take \(P(X)\) to be an adequate representation of your degree of belief in \(X\). Your lower envelope of \(X\) is: \(\underline{P}(X)=\inf P(X)\). Likewise, your upper envelope is \(\overline{P}(X)=\sup P(X)\). They are conjugates of each other in the following sense: \(\overline{P}(X) = 1 - \underline{P}(\neg X)\).