The devil's staircase. Recall the construction of Cantor's ternary set from Problem 7.12. Denote by \(I_{n}^{1}, \ldots, I_{n}^{n}-1\) the intervals which make up \([0,1] \backslash C_{n}\) arranged in increasing order of their endpoints. We construct a sequence of functions \(F_{n}:[0,1] ightarrow[0,1]\) by

\[F_{n}(x):= \begin{cases}0, & \text { if } x=0 \\ i 2^{-n} & \text { if } x \in I_{n}^{i}, 1 \leqslant i \leqslant 2^{n}-1 \\ 1, & \text { if } x=1\end{cases}\]

and interpolate linearly between these values to get \(F_{n}(x)\) for all other \(x\).

(i) Sketch the first three functions \(F_{1}, F_{2}, F_{3}\).

(ii) Show that the limit \(F(x):=\lim _{n ightarrow \infty} F_{n}(x)\) exists.

Remark. \(F\) is usually called the Cantor function.

(iii) Show that \(F\) is continuous and increasing.

(iv) Show that \(F^{\prime}\) exists a.e. and equals 0 .

(v) Show that \(F\) is not absolutely continuous (in the sense of Problem 20.8 (2)) but singular, i.e. the corresponding measure \(\mu\) with distribution function \(F\) is singular w.r.t. Lebesgue measure \(\left.\lambda^{1}ight|_{[0,1]}\).

Data from problem 20.8

 Stielties measure (3). Let \((\mathbb{R}, \mathscr{B}(\mathbb{R}), \mu)\) be a finite measure space and denote by \(F\) the left-continuous distribution function of \(\mu\) as in Problem 6.1. Use Lebesgue's decomposition theorem (Theorem 20.4) to show that we can decompose \(F=F_{1}+F_{2}+F_{3}\) and, accordingly, \(\mu=\mu_{1}+\mu_{2}+\mu_{3}\) in such a way that
Data from problem 7.12

 Cantor's ternary set. Let \((X, \mathscr{A})=([0,1],[0,1] \cap \mathscr{B}(\mathbb{R})), \lambda=\left.\lambda^{1}ight|_{[0,1]}\), and set \(C_{0}=[0,1]\). Remove the open middle third of \(C_{0}\) to get \(C_{1}=J_{1}^{0} \cup J_{1}^{1}\). Remove the open middle thirds of \(J_{1}^{i}, i=0,1\), to get \(C_{2}=J_{2}^{00} \cup J_{2}^{01} \cup J_{2}^{10} \cup I_{2}^{11}\) and so forth.