▁▁▁▁ ▁In ▁other ▁words , ▁each ▁group ▁element ▁ ▁is ▁identified ▁to ▁the ▁evaluation ▁character ▁on ▁the ▁dual . ▁This ▁is ▁strongly ▁analog ous ▁to ▁the ▁canonical ▁isomorphism ▁between ▁a ▁finite - dimensional ▁vector ▁space ▁and ▁its ▁double ▁dual , ▁, ▁while ▁it ▁is ▁worth ▁to ▁point ▁out ▁that ▁any ▁vector ▁space ▁ ▁is ▁an ▁Ab elian ▁group . ▁If ▁ ▁is ▁a ▁finite ▁ab elian ▁group , ▁then ▁ ▁but ▁this ▁isomorphism ▁is ▁not ▁canonical . ▁M aking ▁this ▁statement ▁precise ▁( in ▁general ) ▁requires ▁thinking ▁about ▁dual izing ▁not ▁only ▁on ▁groups , ▁but ▁also ▁on ▁maps ▁ ▁between ▁the ▁groups , ▁in ▁order ▁to ▁treat ▁dual ization ▁as ▁a ▁fun ctor ▁and ▁prove ▁the ▁identity ▁fun ctor ▁and ▁the ▁dual ization ▁fun ctor ▁are ▁not ▁naturally ▁equivalent . ▁Also ▁the ▁du ality ▁theorem ▁implies ▁that ▁for ▁any ▁group ▁( not ▁necessarily ▁finite ) ▁the ▁dual ization ▁fun ctor ▁is ▁an ▁exact ▁fun ctor . ▁ ▁Pon try agin ▁du ality ▁and ▁the ▁Fourier ▁transform ▁ ▁H aar ▁measure ▁ ▁One ▁of ▁the ▁most ▁remarkable ▁facts ▁about ▁a ▁locally ▁compact ▁group ▁G ▁is ▁that ▁it ▁car ries ▁an ▁essentially ▁unique ▁natural ▁measure , ▁the ▁H aar ▁measure , ▁which ▁allows ▁one ▁to ▁consist ently ▁measure ▁the ▁" size " ▁of ▁sufficiently ▁regular ▁subsets ▁ ▁of ▁G . ▁ ▁" Su fficient ly ▁regular ▁subset " ▁here ▁means ▁a ▁Bor el ▁set ; ▁that ▁is , ▁an ▁element ▁of ▁the ▁ σ - algebra ▁generated ▁by ▁the ▁compact ▁sets . ▁More