An equivalent form of uncertain measure

Abstract

A new measure, called uncertain measure, was presented by Liu in 2007 in order to deal with uncertainty, intelligently. The definition of uncertain measure contains Normality axiom, Duality axiom and Subadditivity axiom for any sequence of events. This paper gives an equivalent form of the definition by substituting Subadditivity axiom with a new Subadditivity axiom, Compared with the original axiom, the new axiom only requires that the subadditivity hold for mutually disjoint sets. The equivalent form illustrates that we have twomethods to complete uncertain measure. One method is to preserve original axioms of uncertain measure, and to add the property which is equivalent to the conditions with Normality, Duality and Subadditivity for any sequence of mutually disjoint events. The other is to change the third axiom of uncertain measure into Subadditivity for any sequence of mutually disjoint events, and the front three axioms of uncertain measure is regarded as a property.