Lebesgue-Stieltjes Measure

Proposition (Distribution function is a pre-measure)

Let $F:\mathbb{R}\to \mathbb{R}$ be non-decreasing and Right Continuous. Let $(a_{j},b_{j}],1\le j\le n$ be disjoint h-intervals. (At most one of the $a_{j}$ ’s are allowed to be $-\infty$ and at most one of the $b_{j}$ ’s are allowed to be $+\infty$ , in which case we think of the corresponding interval as $(a_{j},+\infty)$ ). We define $\lambda:\mathscr{A}\to[0,+\infty]$ (where $\mathscr{A}$ is the Collection of all finite disjoint h-intervals) s.t.

$\lambda(\emptyset)=0$
$\lambda\left(\bigsqcup^{n}_{j=1}(a_{j},b_{j}]\right):=\sum_{j=1}^{n}\left(F(b_{j})-F(a_{j})\right)$

Then $\lambda$ is a Pre-measure on $\mathscr{A}$ .

Theorem (Existence of Lebesgue-Stieltjes Measure)

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 $\mu=\mu_{F}$ .

Definition (Lebesgue-Stieltjes measure)

For some non-decreasing and Right Continuous $F:\mathbb{R}\to \mathbb{R}$ , let $\lambda$ be the Pre-measure defined in . Define $\mu^{*}_{F}(E):=\inf\left\{ \sum_{j=1}^{\infty}(F(b_{j})-F(a_{j})):\bigcup_{j=1}^{\infty}(a_{j},b_{j}]\supseteq E \right\}\quad\forall E\subseteq \mathbb{R}$ (which is an Outer Measure) and let $\mathscr{M}_{F}=\{ A\subseteq \mathbb{R}:\mu^{*}_{F}(E)=\mu^{*}_{F}(E\cap A)+\mu_{F}^{*}(E\cap A^{c})\ \forall E\subseteq \mathbb{R} \}$ By Carathéodory Theorem, we have that $\mathscr{M}_{F}$ is a σ-algebra and setting $\left. \mu_{F}^{*}\right|_{\mathscr{M}_{F}}=\bar{\mu}_{F}$ we have that $(\mathbb{R},\mathscr{M}_{F},\bar{\mu}_{F})$ is a Complete Measure Space. We call $\bar{\mu}_{F}$ the Lebesgue-Stieltjes measure. Also, noting that $\mathcal{B}\subset \mathscr{M}_{F}$ We have that $\mu_{F}=\left. \bar{\mu}_{F}\right|_{\mathcal{B}}$ is a Borel measure.

Lemma (Lebesgue-Stieltjes on Open intervals)

For $\mathscr{M}\subset 2^{\mathbb{R}}$ : $\forall E\in\mathscr{M}: \mu_{F}(E)=\inf\left\{ \sum_{j=1}^{\infty}\mu((a_{j},b_{j})):\bigcup_{j=1}^{\infty}(a_{j},b_{j})\supseteq E \right\}=:\nu(E)$

Theorem (Lebesgue-Stieltjes Measures are Regular)

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

Definition ( $G_\delta$ set)

Let $(X,\mathscr{T})$ be a topological space. Let $\mathcal{B}$ be the Borel σ-algebra associated with $\mathscr{T}$ . Then we say $G_{\delta}\text{ sets}:=\bigcap_{n\in\mathbb{N}}A_{n}\quad\forall(A_{n})_{n\in\mathbb{N}}\subset \mathscr{T}$ or $\text{Countable intersections of open sets}:=G_{\delta}\text{ sets}$

Example

$\mathbb{R}\setminus \mathbb{Q}\in\mathcal{B}(G_{\delta})$

Definition ( $F_\sigma$ set)

Let $(X,\mathscr{T})$ be a topological space. Let $\mathcal{B}$ be the Borel σ-algebra associated with $\mathscr{T}$ . Then we say $F_{\sigma}:=\bigcup_{n\in\mathbb{N}}C_{n}\quad\forall(C_{n})_{n\in\mathbb{N}}\text{ s.t. }C_{n}^{c}\in\mathscr{T},\ \forall n\in\mathbb{N}$ or $\text{Countable unions of }\textbf{CLOSED} \text{ sets}:=F_{\sigma}\text{ sets}$

Example

$\mathbb{Q}\in\mathcal{B}(F_{\sigma})$

Theorem (Simplicity of Lebesgue-Stieltjes Measurable Sets)

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:

$E\in\mathscr{M}$
$E=V\setminus N_{1}$ , where $V$ is a Gδ set and $\mu(N_{1})=0$
$E=H\cup N_{2}$ , where $H$ is an Fσ set and $\mu(N_{2})=0$

Definition (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 $g\text{ increasing}\implies g(c^{+}) \ \ \& \ \ g(c^{-})\text{ exist }\forall c\in\mathbb{R}^{+}$

Proposition

The Lebesgue-Stieltjes Measure, $\mu_{g}$ extends to a σ-finite measure on the Borel σ-algebra, $\mathcal{B}(\mathbb{R}^{+})$ .

Theorem

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 $\begin{gather}g\text{ is the difference of two increasing functions}\\\iff \\ g\text{ is of finite variation on }\mathbb{R}^{+}\end{gather}$

Lebesgue-Stieltjes Measure

437

Linked from