\begin{enumerate}[label=(\roman*)] \item If $H: {\bf R} \rightarrow {\bf R}$ is the Heaviside function $H := 1_{[0,+\infty)}$, show that $\mu_H$ is equal to the Dirac measure $\delta_0$ at the origin (defined in Example 1.4.22). \item If $F = \sum_n c_n J_n$ is a jump function (as defined in Definition 1.6.30), show that $\mu_F$ is equal to the linear combination $\sum c_n \delta_{x_n}$ of Dirac measures (as defined in \href{./exercise_1-4-22}{Exercise 1.4.22}), where $x_n$ is the point of discontinuity for the basic jump function $J_n$. \end{enumerate}

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Exercise 1.7.14 (Lebesgue-Stieltjes measure, pure point case)

(i): If $H : R \to R$ is the Heaviside function $H : = 1_{[0, + \infty)}$ , show that $μ_{H}$ is equal to the Dirac measure $δ_{0}$ at the origin (defined in Example 1.4.22).
(ii): If $F = \sum_{n} c_{n} J_{n}$ is a jump function (as defined in Definition 1.6.30), show that $μ_{F}$ is equal to the linear combination $\sum c_{n} δ_{x_{n}}$ of Dirac measures (as defined in Exercise 1.4.22), where $x_{n}$ is the point of discontinuity for the basic jump function $J_{n}$ .