D Solutions and hints to selected exercises

Then y = (x1,x2, . . .) is as desired. The sum, x = ∑∞k=1 xk/2k, is uniformly distributed over [0,1]: its distribution function F(u) = u for each x that has a finite dyadic expansion, u = ∑Nk=1 εk/2k, with εk = 0 or 1. Since any distribution function is non-decreasing, F(u) = u for all u ∈ [0,1].