A Short Proof of the Transcendence of Thue-Morse Continued Fractions

a finite measure. By the Lebesgue decomposition theorem that measure is the sum of a measure ?jl such that ?jl <? X and a measure v for which there is a Lebesgue null set A contained in 7 with v(J \ A) = 0. It follows from Theorem 6.3 that the cumu lative distribution function Fv has a zero derivative Lebesgue almost everywhere in 7. Since F? has a derivative Lebesgue almost everywhere in 7, the same is true for F = F? + Fv. Thus, with the inequality proved as in [7, p. 101], we have the following result: