▁take ▁ ▁to ▁be ▁the ▁push forward ▁of ▁ ▁under ▁the ▁inverse ▁map ▁on ▁, ▁namely ▁ ▁for ▁any ▁Bor el ▁set ▁. ▁This ▁complex ▁measure ▁has ▁Fourier ▁coefficients ▁. ▁We ▁are ▁going ▁to ▁apply ▁the ▁above ▁to ▁the ▁convolution ▁between ▁ ▁and ▁, ▁namely ▁we ▁choose ▁, ▁meaning ▁that ▁ ▁is ▁the ▁push forward ▁of ▁the ▁measure ▁ ▁( on ▁) ▁under ▁the ▁product ▁map ▁. ▁By ▁F ub ini ' s ▁theorem ▁ ▁So , ▁by ▁the ▁identity ▁derived ▁earlier , ▁ ▁By ▁F ub ini ' s ▁theorem ▁again , ▁the ▁right - hand ▁side ▁equals ▁▁ ▁The ▁proof ▁of ▁the ▁analog ous ▁statement ▁for ▁the ▁real ▁line ▁is ▁identical , ▁except ▁that ▁we ▁use ▁the ▁identity ▁ ▁( which ▁follows ▁from ▁F ub ini ' s ▁theorem ), ▁where ▁. ▁We ▁observe ▁that ▁, ▁ ▁and ▁ ▁for ▁, ▁which ▁converges ▁to ▁ ▁as ▁. ▁So , ▁by ▁domin ated ▁convergence , ▁we ▁have ▁the ▁analog ous ▁identity ▁ ▁Con sequ ences ▁ ▁A ▁real ▁or ▁complex ▁Bor el ▁measure ▁ ▁on ▁the ▁circle ▁is ▁diff use ▁( i . e . ▁) ▁if ▁and ▁only ▁if ▁. ▁ ▁A ▁probability ▁measure ▁ ▁on ▁the ▁circle ▁is ▁a ▁Di rac ▁mass ▁if ▁and ▁only ▁if ▁. ▁( Here , ▁the ▁n ont rivial ▁imp lication ▁follows ▁from ▁the ▁fact ▁that ▁the ▁weights ▁ ▁are ▁positive ▁and ▁satisfy ▁, ▁which ▁forces ▁ ▁and ▁thus ▁, ▁so ▁that ▁there ▁must ▁be ▁a ▁single ▁atom ▁with ▁mass ▁. ) ▁ ▁References ▁