Counterexample. Let X be a random variable taking on the values −1, 0, 1, 2, . . . with probabilities Pθ{X = −1} = θ; Pθ{X = x} = (1 − θ)

2

θx, x = 0, 1,....

Then P = {Pθ, 0 <θ< 1} is boundedly complete but not complete. [Girschick et al. (1946)]