The answer is in this paper :: Yuji Takahashi, Functions with a unique mean value and amenability. Proc. Amer. Math. Soc. 121 (1994), 775-777 (http://www.ams.org/journals/proc/1994-121-03/S0002-9939-1994-1189549-1/home.htmlFreely accessible here on AMS' site)
