14. Measurability of testsof Theorer: 3. The function l/J3 defined by (16) and (17) is jointly measurable in u and t. [With C1 = v and C2 = w, the determining equations for v. W , 'YI ' 'Y2 are (25) F,( v -) + [1 - F,( w)] + 'YI [ F,( v) - F, (v - )] +'Y2[F,(W) - F,(w -)] = a and (26) G,(v - ) + [1 - G, ( w)] + 'YI [ G,( v) - G,(v - )] +'Y2[G,( w) - G,( w -)] =

a. where (27) F,(u) = r C,(81) e81Y dJl,(y), -00 G,(u) = r C,(82)e82Y dJl,(y) -00 denote the conditional cumulative distribution function of U given t when 8 = 81 and 8 = 82 respectively.

(1) For each 0 y a let v(y, t) = F;1(y) and w(y, t) = F;-I(1 - a + y), where the inverse function is defined as in the proof of Theorem 3. Define 'YI (y, t) and 'Y2 (y, t) so that for v = v(y, t) and w = w(y, t), F;( v -) + 'YI [F;(v) - F;( v - )] = y, 1- F;(w) + 'Y2[F;(w) - F;(w - )] = a - y. (2) Let H(y, t) denote the left-hand side of (26), with v = v(y, r), etc. Then H(O, t) > a and Hi«, t) <

a. This follows by Theorem 2 of Chapter 3 from the fact that v(O, t) = - 00 and w(

a, t) = 00 (which shows the conditional tests corresponding to y = 0 and y = a to be one-sided), and that the left-hand side-of (26) for any y is the power of this conditional test. (3) For fixed t, the functions HI ( y , t) = G,(v -) + 'YI [ G,( v) - G,( v - )] and H2(y, t) = 1 - G,(w) + 'Y2[ G,(w) - G,(w -)] are continuous functions of y. This is a consequenceof the fact, which follows from (27), that a.e. fJJT the discontinuities and flat stretches of F; and G, coincide. (4) The function H(y, t) is jointly measurable in y and t. This follows from the continuity of H by an argument similar to the proof of measurability of F; ( u) in the text. Define y(t) = inf{y : H(y,t) < a}, and let v(t) = v[y(t), t], etc. Then (25) and (26) are satisfied for all t. The measurability of v(t), w(t), 'YI (r), and 'Y2(t) defined in this manner will follow from measurability in t of y(t) and F;1[y(t)]. This is a consequence of the relations, which hold for all real

c, {t :y(t)