=+26 *7. 25.131 The uniqueness theorem for characteristic functions can be derived from the Weierstrass approximation theorem. Fill in the details of the follow-

ing argument. Let u and v be probability measures on the line. For continuous f with bounded support choose a so that u(-a,

a) and v(-a,

a) are nearly 1 and f vanishes outside (-a, a). Let g be periodic and agree with f in (-a,a), and by the Weierstrass theorem uniformly approximate g(x) by a trigonomet-

ric sum p(x) = C_jake" **. If u and v have the same characteristic function, then ffdu = fgdu = [pdu = [pdv = fgdv = ffdv.