▁precisely , ▁ ▁a ▁right ▁H aar ▁measure ▁on ▁a ▁locally ▁compact ▁group ▁G ▁is ▁a ▁count ably ▁add itive ▁measure ▁ μ ▁defined ▁on ▁the ▁Bor el ▁sets ▁of ▁G ▁which ▁is ▁right ▁invariant ▁in ▁the ▁sense ▁that ▁ μ ( A x ) ▁= ▁ μ ( A ) ▁for ▁ ▁x ▁an ▁element ▁of ▁G ▁and ▁A ▁a ▁Bor el ▁subset ▁of ▁G ▁and ▁ ▁also ▁satisfies ▁some ▁regular ity ▁conditions ▁( sp elled ▁out ▁in ▁detail ▁in ▁the ▁article ▁on ▁H aar ▁measure ). ▁Ex cept ▁for ▁positive ▁scaling ▁factors , ▁a ▁H aar ▁measure ▁on ▁G ▁is ▁unique . ▁ ▁The ▁H aar ▁measure ▁on ▁G ▁allows ▁us ▁to ▁define ▁the ▁notion ▁of ▁integral ▁for ▁( complex - valu ed ) ▁Bor el ▁functions ▁defined ▁on ▁the ▁group . ▁ ▁In ▁particular , ▁one ▁may ▁consider ▁various ▁L p ▁spaces ▁associated ▁to ▁the ▁H aar ▁measure ▁ μ . ▁Specifically , ▁ ▁Note ▁that , ▁since ▁any ▁two ▁H aar ▁measures ▁on ▁G ▁are ▁equal ▁up ▁to ▁a ▁scaling ▁factor , ▁this ▁L p - space ▁is ▁independent ▁of ▁the ▁choice ▁ ▁of ▁H aar ▁measure ▁and ▁thus ▁perhaps ▁could ▁be ▁written ▁as ▁L p ( G ). ▁ ▁However , ▁the ▁L p - norm ▁on ▁this ▁space ▁depends ▁on ▁the ▁choice ▁of ▁H aar ▁measure , ▁so ▁if ▁one ▁wants ▁to ▁talk ▁about ▁is omet ries ▁it ▁is ▁important ▁to ▁keep ▁track ▁of ▁the ▁H aar ▁measure ▁being ▁used . ▁ ▁Fourier ▁transform ▁and ▁Fourier ▁in