A Note on Extensions of Non-additive Measures

The motivation for our consideration comes from the fuzzy set theory in a potential relation to quantum theories and mathematical economics: Given a certain non-additive assignment on a Boolean algebra (a kind of "belief measure"), can this assignment be extended over a larger Boolean algebra? We answer this question in the affirmative. By examining the universality of the method used, we conclude that even when we let the assignment subject to an arbitrary collection of unsharp inequalities, we are always able to extend the measure so defined over a larger Boolean algebra.