\begin{enumerate}[label=(\roman*)] \item If $F: {\bf R} \rightarrow {\bf R}$ is a monotone non-decreasing function, show that $F$ is continuous if and only if $\mu_F(\{x\})=0$ for all $x \in {\bf R}$. \item If $F$ is the Cantor function (defined in \href{./exercise_1-6-48}{Exercise 1.6.48}), show that $\mu_F$ is a probability measure supported on the middle-thirds Cantor set (\href{./exercise_1-2-9}{Exercise 1.2.9}) in the sense that $\mu_F({\bf R} \backslash C) = 0$. The measure $\mu_F$ is known as Cantor measure. \item If $\mu_F$ is Cantor measure, establish the self-similarity properties $\mu( \frac{1}{3} \cdot E ) = \frac{1}{2} \mu(E)$ and $\mu( \frac{1}{3} \cdot E + \frac{2}{3} ) = \frac{1}{2} \mu(E)$ for every Borel-measurable $E \subset [0,1]$, where $\frac{1}{3} \cdot E := \{ \frac{1}{3} x: x \in E \}$. \end{enumerate}

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Exercise 1.7.15 (Lebesgue-Stieltjes measure, singular continuous case)

(i): If $F : R \to R$ is a monotone non-decreasing function, show that $F$ is continuous if and only if $μ_{F} ({x}) = 0$ for all $x \in R$ .
(ii): If $F$ is the Cantor function (defined in Exercise 1.6.48), show that $μ_{F}$ is a probability measure supported on the middle-thirds Cantor set (Exercise 1.2.9) in the sense that $μ_{F} (R ∖ C) = 0$ . The measure $μ_{F}$ is known as Cantor measure.
(iii): If $μ_{F}$ is Cantor measure, establish the self-similarity properties $μ (\frac{1}{3} \cdot E) = \frac{1}{2} μ (E)$ and $μ (\frac{1}{3} \cdot E + \frac{2}{3}) = \frac{1}{2} μ (E)$ for every Borel-measurable $E \subset [0, 1]$ , where $\frac{1}{3} \cdot E : = {\frac{1}{3} x : x \in E}$ .