▁Let ▁X ▁be ▁the ▁universe : ▁the ▁set ▁representing ▁all ▁possible ▁states ▁of ▁a ▁system ▁under ▁consideration . ▁The ▁power ▁set ▁ ▁is ▁the ▁set ▁of ▁all ▁subsets ▁of ▁X , ▁including ▁the ▁empty ▁set   . ▁For ▁example , ▁if : ▁ ▁then ▁ ▁The ▁elements ▁of ▁the ▁power ▁set ▁can ▁be ▁taken ▁to ▁represent ▁propos itions ▁concerning ▁the ▁actual ▁state ▁of ▁the ▁system , ▁by ▁containing ▁all ▁and ▁only ▁the ▁states ▁in ▁which ▁the ▁proposition ▁is ▁true . ▁ ▁The ▁theory ▁of ▁evidence ▁assign s ▁a ▁belief ▁mass ▁to ▁each ▁element ▁of ▁the ▁power ▁set . ▁Form ally , ▁a ▁function ▁ ▁is ▁called ▁a ▁basic ▁belief ▁assignment ▁( B BA ), ▁when ▁it ▁has ▁two ▁properties . ▁First , ▁the ▁mass ▁of ▁the ▁empty ▁set ▁is ▁zero : ▁ ▁Second , ▁the ▁masses ▁of ▁the ▁remaining ▁members ▁of ▁the ▁power ▁set ▁add ▁up ▁to ▁a ▁total ▁of ▁ 1 : ▁ ▁The ▁mass ▁m ( A ) ▁of ▁A , ▁a ▁given ▁member ▁of ▁the ▁power ▁set , ▁express es ▁the ▁proportion ▁of ▁all ▁relevant ▁and ▁available ▁evidence ▁that ▁supports ▁the ▁claim ▁that ▁the ▁actual ▁state ▁belongs ▁to ▁A ▁but ▁to ▁no ▁particular ▁subset ▁of ▁A . ▁The ▁value ▁of ▁m ( A ) ▁per tain s ▁only ▁to ▁the ▁set ▁A ▁and ▁makes ▁no ▁additional ▁claims ▁about ▁any ▁subsets ▁of ▁A , ▁each ▁of ▁which ▁have , ▁by ▁definition , ▁their ▁own ▁mass . ▁ ▁From ▁the ▁mass ▁assign ments , ▁the ▁upper ▁and ▁lower ▁bounds ▁of ▁a ▁probability