Let $F: {\bf R} \rightarrow {\bf R}$ be a monotone non-decreasing function, and define the left and right limits $$\displaystyle F_-(x) := \sup_{y x} F(y),$$ Let ${\mathcal B}[{\bf R}]$ be the Borel $\sigma$-algebra on ${\bf R}$. Then there exists a unique Borel measure $\mu_F: {\mathcal B}[{\bf R}] \rightarrow [0,+\infty]$ such that \begin{align} \displaystyle \mu_F( [a,b] ) &= F_+(b)-F_-(a),\quad \mu_F( [a,b) ) = F_-(b)-F_-(a),\\ \displaystyle \mu_F( (a,b] ) &= F_+(b)-F_+(a),\quad \mu_F( (a,b) ) = F_-(b) - F_+(a) \end{align} for all $-\infty < b < a < \infty$, and $$\displaystyle \mu_F( \{a\} ) = F_+(a) - F_-(a)$$ for all $a \in {\bf R}$.

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Exercise 1.7.10 (Existence of Lebesgue-Stieltjes measure)

Let $F : R \to R$ be a monotone non-decreasing function, and define the left and right limits

F_{−} (x) : = \sup_{y < x} F (y); F_{+} (x) : = \inf_{y > x} F (y),

Let $ℬ [R]$ be the Borel $σ$ -algebra on $R$ . Then there exists a unique Borel measure $μ_{F} : ℬ [R] \to [0, + \infty]$ such that

\begin{align} μ_{F} ([a, b]) & = F_{+} (b) - F_{−} (a), μ_{F} ([a, b)) = F_{−} (b) - F_{−} (a), & (1) \\ μ_{F} ((a, b]) & = F_{+} (b) - F_{+} (a), μ_{F} ((a, b)) = F_{−} (b) - F_{+} (a) & (2) \end{align}

for all $- \infty < b < a < \infty$ , and

μ_{F} ({a}) = F_{+} (a) - F_{−} (a)

for all $a \in R$ .