The following example shows that Corollary 3.6.1 does not extend to a countably infinite family of distributions. Let pn be the uniform probability density on [0, 1 + 1/n], and p0 the uniform density on (0, 1).

(i) Then p0 is linearly independent of (p1, p2, . . .), that is, there do not exist constants c1, c2,... such that p0 = cn pn.

(ii) There does not exist a test φ such that φpn = α for n = 1, 2,... but

φp0 > α.