Exercise 6, § 6

a) Let K be the Cantor triadic set in the interval [0, 1] (GT, IV, §2, No. 5). Show that there exist a diffuse positive measure $\nu$ on $\mathbf{R}$, with support K and total mass 1, and a continuous increasing function $\theta$, defined on $\mathbf{R}$, such that $\theta(\mathbf{R}) = I$ and $\theta(\nu) = \varphi_I \cdot \mu$ ($\mu$ the Lebesgue measure; cf. GT, IV, §8, Exer. 16).

b) Deduce from a) that there exists a diffuse positive measure on $\mathbf{R}$, alien to Lebesgue measure, whose support is all of I (in each interval J contiguous to K and contained in I, take a measure proportional to the image of $\nu$ under an affine mapping of I onto J; then proceed by recursion).