51. Suppose that a critical function 1/10 satisfies (65) but not (67), and let a < t. Then the following construction provides a measurable critical function 1/1 satisfying (67) and such that I/Io(z) I/I( z) for all z. Inductively, sequences of functions 1/11 '1/12' ... and "'0 ' "'I' ... are defined through the relations and "'m(Z) = [ I/Im(Z') z' e S(z ) NI ! · · · Nc', ' m=O,I, ... , { I/Im- I( Z) + [a - "'m_I(Z)] I/Im(z) = if both I/Im -I (z) and "'m-I(z) are I/Im -I (z) otherwise. <

a, The function 1/1 (z) = lim I/Im(z) then satisfies the required condi tions. [The functions I/Imare nondecreasing and between 0 and 1. It is further seen by induction that 0 a - "'m(z) (1 - y)m[a - "'o(z»), where y = 1/ NI! · ··N).)