12. The completeness of the order statistics in Example 6 remains true if the family $' is replaced by the family of all continuous distributions. [To show that for any integrable symmetric function l/J. !l/J(x1,... , x,,) dF(xl ) dF(x,,) = 0 for all continuous F implies l/J = 0 a.e., replace F by al F1 + +a" F,., where 0 <

a, < 1. Eaj = 1. By considering the left side of the resulting identity as a polynomial in the a's one sees that !l/J(XI" ' " x,,) dFI(XI) ' " dF,.(x,,) = 0 for all continuous F; . This last equation remains valid if the F; are replaced by la ( x) F( x ), where la ( x) = 1 if X:$;

a, and = 0 otherwise. This implies that l/J'= 0 except on a set which has measure 0 under F X .• . X F for all continuous F.)