As a unified generalization of both the Lebesgue integral and the Suge
no integral, pan-integrals were introduced for nonnegative measurable
functions with respect to fuzzy measures. In this paper, the concept o
f a pan-integral is generalized for nonnegative measurable functions w
ith respect to monotone set functions, which include coherent imprecis
e probabilities. An inequality is established between two functionals
on the class of all nonnegative bounded measurable functions, the natu
ral extension of a monotone lower probability and the corresponding pa
n-integral whose operators are the common addition and common multipli
cation. We also discuss the necessary and sufficient conditions for th
e equality between these two functionals. By using a new concept of pa
ra-additivity, we derive a necessary and sufficient condition for the
equality when the lower probability is restricted to be a belief measu
re.