Continuation. The function 4 defined by (4.16), (4.18), and (4.19) is jointly measurable in u and t.

Continuation. The function φ4 defined by (4.16), (4.18), and (4.19)

is jointly measurable in u and t.

[The proof, which otherwise is essentially like that outlined in the preceding problem, requires the measurability in z and t of the integral g(z, t) =

z−

−∞

udFt(u).

This integral is absolutely convergent for all t, since Ft is a distribution belonging to an exponential family. For any z < ∞, g(z, t) = lim gn(z, t), where gn(z, t) = ∞

j=1



z − j 2n

 Ft



z − j − 1 2n − 0



− Ft



z − j 2n − 0

 ,

and the measurability of g follows from that of the functions gn. The inequalities corresponding to those obtained in step (2) of the preceding problem result from the property of the conditional one-sided tests established in Problem 3.50.]