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) dF(x1) ...

dF(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 the α’s one sees that φ(x1,...,xn) dF1(x1) . . . dFn(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.]