Lower and upper values

Given a convex set K of probability distributions, a probability interval can be created for every event A by defining lower and upper bounds:

$$\underline{p}(A) = \inf_{p \in K} p(A), \hspace{1cm} \overline{p}(A) = \sup_{p \in K} p(A).$$

Lower and upper expectations for a function f(X) are defined as (E_p[f] is the expectation of the function f):

$$\underline{E}[f] = \inf_{p \in K} E_p[f], \hspace{1cm} \overline{E}[f] = \sup_{p \in K} E_p[f].$$

There is a one-to-one correspondence between lower (or upper) expectations and credal sets. Given a credal set, the set of all lower expectations for all arbitrary functions f(X) is unique, and vice-versa.