The two sides of a fourier-stieltjes transform and almost idempotent measures

For measures μ on the circle T the quantities {Mathematical expression}, {Mathematical expression} need not be equal; it is shown, however, that they are continuous with respect to each other when μ varies on bounded subsets of M(T), the space of measures on T. It is also shown that measures μ which are e{open}-almost idempotent (i.e. {Mathematical expression}) are the sum of an idempotent measure and of a measure υ satisfying {Mathematical expression} provided e{open} is small enough (as a function of {norm of matrix}μ{norm of matrix}).