(Lebesgue-Stieltjes measure, pure point case). (i) If H: 𝐑 →𝐑 is the Heaviside function H:=1[0,+

(Lebesgue-Stieltjes measure, pure point case). (i) If H: 𝐑...

(Lebesgue-Stieltjes measure, pure point case). (i) If $H: \mathbf{R} \rightarrow \mathbf{R}$ 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). (ii) 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 delta functions (as defined in Exercise 1.4.22), where $x_n$ is the point of discontinuity for the basic jump function $J_n$.

(Lebesgue-Stieltjes measure, pure point case).
(i) If $H: \mathbf{R} \rightarrow \mathbf{R}$ 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).
(ii) 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 delta functions (as defined in Exercise 1.4.22), where $x_n$ is the point of discontinuity for the basic jump function $J_n$.

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