The completeness of the order statistics in Example 4.3.4 remains true if the family F is replaced by the family F1 of all continuous distributions.

[Due to Fraser (1956). To show that for any integrable symmetric function φ,

φ(x1,..., xn) d F(x1)...

d F(xn) = 0 for all continuous F implies φ = 0 a.e., replace F by α1F1 +···+

αnFn, where 0 < αi < 1,

αi = 1. By considering the left side of the resulting identity as a polynomial in theα’s one sees that

φ(x1,..., xn) d F1(x1)... d Fn(xn) =

0 for all continuous Fi . This last equation remains valid if the Fi are replaced by Iai(x)F(x), where Iai(x) = 1 if x ≤ ai and = 0 otherwise. This implies that φ = 0 except on a set which has measure 0 under F × ... × F for all continuous F.]