Define a Radon measure on ${\bf R}$ to be a Borel measure $\mu$ obeying the following additional properties: \begin{enumerate}[label=(\roman*)] \item (Local finiteness) $\mu(K) < \infty$ for every compact $K$. \item (Inner regularity) One has $\mu(E) = \sup_{K \subset E, K \hbox{ compact}} \mu(K)$ for every Borel set $E$. \item (Outer regularity) One has $\mu(E) = \inf_{U \supset E, U \hbox{ open}} \mu(U)$ for every Borel set $E$. \end{enumerate} Show that for every monotone function $F: {\bf R} \rightarrow {\bf R}$, the Lebesgue-Stieltjes measure $\mu_F$ is a Radon measure on ${\bf R}$; conversely, if $\mu$ is a Radon measure on ${\bf R}$, show that there exists a monotone function $F: {\bf R} \rightarrow {\bf R}$ such that $\mu= \mu_F$.

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Exercise 1.7.11 (Radon measure)

Define a Radon measure on $R$ to be a Borel measure $μ$ obeying the following additional properties:

(i): (Local finiteness) $μ (K) < \infty$ for every compact $K$ .
(ii): (Inner regularity) One has $μ (E) = \sup_{K \subset E, K compact} μ (K)$ for every Borel set $E$ .
(iii): (Outer regularity) One has $μ (E) = \inf_{U \supset E, U open} μ (U)$ for every Borel set $E$ .

Show that for every monotone function $F : R \to R$ , the Lebesgue-Stieltjes measure $μ_{F}$ is a Radon measure on $R$ ; conversely, if $μ$ is a Radon measure on $R$ , show that there exists a monotone function $F : R \to R$ such that $μ = μ_{F}$ .

Exercise 1.7.11 (Radon measure)

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