It is shown that the integral Hausdorff mean Tp of the Fourier-Stieltjes transform of a measure on the real line is the Fourier transform of an L1 function if and only if Tp. vanishes at infinity or the kernel of T has mean value zero. Also a sufficient condition on the kernel of T and a necessary and sufficient condition on the measure is established in order for-i sï%r{x)T p(x) to be the Fourier transform of an L1-function. These results yield an improvement of Fejer's and Wiener's formulas for the inversion of Fourier-Stieltjes transforms, the uniqueness property of certain generalized Fourier transforms, and a generalization of the mean ergodic theorem for unitary operators. Let Af (/?) be the space of complex bounded regular Borel measures on the real line R and ßix) be the Fourier-Stieltjes transform of a measure p in MiR), pix)= [ e~'xt dpit), xeR. Jr We consider the integral Hausdorff mean Tp generated by a Borel measurable kernel ip in LliR), which is defined for x e R by Tpix)= / fiixs)y/is)ds =-/ tp (?-) piy)dy, Jr \x\ Jr xx/