▁belief ▁functions ▁measured ▁with ▁capac ities ▁is ▁a ▁way ▁to ▁recon cile ▁the ▁Ell s berg ▁par ado x ▁and ▁the ▁Al lais ▁par ado x . ▁ ▁Definition ▁The ▁following ▁notation ▁is ▁used : ▁▁▁ ▁- ▁a ▁set . ▁▁ ▁- ▁a ▁collection ▁of ▁subsets ▁of ▁. ▁▁▁ ▁- ▁a ▁function . ▁▁ ▁- ▁a ▁monot one ▁set ▁function . ▁ ▁Assume ▁that ▁ ▁is ▁meas urable ▁with ▁respect ▁to ▁, ▁that ▁is ▁ ▁Then ▁the ▁Cho quet ▁integral ▁of ▁ ▁with ▁respect ▁to ▁ ▁is ▁defined ▁by : ▁ ▁where ▁the ▁integr als ▁on ▁the ▁right - hand ▁side ▁are ▁the ▁usual ▁Riemann ▁integral ▁( the ▁integr ands ▁are ▁integra ble ▁because ▁they ▁are ▁monot one ▁in ▁). ▁ ▁Properties ▁ ▁In ▁general ▁the ▁Cho quet ▁integral ▁does ▁not ▁satisfy ▁add itivity . ▁More ▁specifically , ▁if ▁ ▁is ▁not ▁a ▁probability ▁measure , ▁it ▁may ▁hold ▁that ▁ ▁for ▁some ▁functions ▁ ▁and ▁. ▁ ▁The ▁Cho quet ▁integral ▁does ▁satisfy ▁the ▁following ▁properties . ▁ ▁Mon ot onic ity ▁If ▁ ▁then ▁ ▁Pos itive ▁hom ogene ity ▁ ▁For ▁all ▁ ▁it ▁holds ▁that ▁ ▁Com on ot one ▁add itivity ▁ ▁If ▁ ▁are ▁com on ot one ▁functions , ▁that ▁is , ▁if ▁for ▁all ▁ ▁it ▁holds ▁that ▁. ▁which ▁can ▁be ▁thought ▁of ▁as ▁ ▁and ▁ ▁rising ▁and ▁falling ▁together ▁ ▁then ▁ ▁Sub add itivity ▁ ▁If ▁ ▁is ▁ 2 - altern ating , ▁then ▁ ▁Super add itivity ▁ ▁If