(Lebesgue-Stieltjes measure, singular continuous case).
(i) If $F: \mathbf{R} \rightarrow \mathbf{R}$ is a monotone non-decreasing function, show that $F$ is continuous if and only if $\mu_F(\{x\})=0$ for all $x \in \mathbf{R}$.
(ii) If $F$ is the Cantor function (defined in Exercise 1.6.47), show that $\mu_F$ is a probability measure supported on the middle-thirds Cantor set (see Exercise 1.2.9) in the sense that $\mu_F(\mathbf{R} \backslash C)=0$. The measure $\mu_F$ is known as Cantor measure.
(iii) If $\mu_F$ is Cantor measure, establish the self-similarity properties $\mu\left(\frac{1}{3} \cdot E\right)=\frac{1}{2} \mu(E)$ and $\mu\left(\frac{1}{3} \cdot E+\frac{2}{3}\right)=\frac{1}{2} \mu(E)$ for every Borel-measurable $E \subset[0,1]$, where $\frac{1}{3} \cdot E:=\left\{\frac{1}{3} x: x \in E\right\}$.