Let $F:\mathbb{R}\to \mathbb{R}$ be non-decreasing and right continuous. Then there exists a unique Borel measure $\mu_{F}$ such that $$\mu_{F}((a,b])=F(b)-F(a)\quad\forall a<b$$Conversely, if $\mu$ is a Borel measure on $\mathbb{R}$ that is finite on all bounded Borel subsets, then defining $$F(x)=\begin{cases} \mu((0,x]) & x>0 \\ 0 & x=0 \\ -\mu((x,0]) & x<0 \end{cases}$$ we have that $F$ is increasing and right continuous and $\mu=\mu_{F}$.

Let $\bar{\mu}_{F}$ be the complete Lebesgue-Stieltjes Measure on $\mathscr{M}_{F}$. Then $\forall E\in\mathscr{M}$ we have $$\begin{align*} \bar{\mu}_{F}&=\inf\{ \bar{\mu}_{F}(U):U\text{ open, }U\supseteq E \}\\ \bar{\mu}_{F}(E)&=\sup \{ \bar{\mu}_{F}(K):K\text{ compact, }K\subseteq E \} \end{align*}$$

Let $\overline{\mu}_{F}:\mathscr{M}\to[0,+\infty]$ be a complete Lebesgue-Stieltjes Measure, where the σ-algebra $\mathscr{M}=\mathscr{M}_{F}$. Then the following are equivalent: 1. $E\in\mathscr{M}$ 2. $E=V\setminus N_{1}$, where $V$ is a Gδ set and $\mu(N_{1})=0$ 3. $E=H\cup N_{2}$, where $H$ is an Fσ set and $\mu(N_{2})=0$ # 437 [!def] 437 definition Let $g:\mathbb{R}\to \mathbb{R}^{+}$ be an increasing function. Define $\mu_{g}$ on the set of all bounded intervals of $\mathbb{R}^{+}$ by $$\begin{align*} \mu_{g}([a,b])=g(b^{+})-g(a^{-})\\ \mu_{g}(]a,b])=g(b^{+})-g(a^{+})\\ \mu_{g}([a,b[)=g(b^{-})-g(a^{-})\\ \mu_{g}(]a,b[)=g(b^{-})-g(a^{+})\\ \end{align*} \ \text{ where }0\le a\le b$$and $$\begin{align*} g(c^{+})=\lim_{ x \to c \ x>c }g(x)\\ g(c^{-})=\lim_{ x \to c, x<c } g(x) \end{align*}$$where $$g\text{ increasing}\implies g(c^{+}) \ \ \& \ \ g(c^{-})\text{ exist }\forall c\in\mathbb{R}^{+}$$

Let $g:\mathbb{R}^{+}\to \mathbb{R}$. Then: $$g=g_{1}-g_{2}\text{ where }g_{1},g_{2}:\mathbb{R}^{+}\to \mathbb{R}\text{ increasing }\iff g\text{ is of finite variation on }\mathbb{R}^{+}$$or \begin{array} -g\text{ is the difference of two increasing functions}\\ \iff \\ g\text{ is of finite variation on }\mathbb{R}^{+} \end{array}