Coherent updating of non-additive measures

• We study the natural and regular extensions of a 2-monotone lower prevision.
• Their equality means the uniqueness of the updated models that preserve coherence.
• We characterise this equality, and study some particular cases.
• For ∞-monotone lower previsions we express it in terms of the Möbius inverse.
• For minimum-preserving ones we link it to updating rules for possibility measures.

The conditions under which a 2-monotone lower prevision can be uniquely updated (in the sense of focusing) to a conditional lower prevision are determined. Then a number of particular cases are investigated: completely monotone lower previsions, for which equivalent conditions in terms of the focal elements of the associated belief function are established; random sets, for which some conditions in terms of the measurable selections can be given; and minitive lower previsions, which are shown to correspond to the particular case of vacuous lower previsions.