Stieltjesscher content

The Stieltjesian content , named after the mathematician Thomas Jean Stieltjes , is a content that can be used to generalize the Riemann integral to the Lebesgue-Stieltjes integral .

Stieltjesian content

The Stieltjes content is defined on the half-ring via . Since one can continue content on a half-ring clearly on its generated ring , it can on the set ${\displaystyle {\mathcal {J}}:=\{]a,b]:a,b\in \mathbb {R} ,a\leq b\}}$ ${\displaystyle \mathbb {R} }$

${\displaystyle {\mathcal {F}}:=\left\{\left.\bigcup _{j=1}^{n}I_{j}\,\right|\,I_{1},\dotsc ,I_{n}\in {\mathcal {J}},I_{j}\cap I_{k}=\emptyset {\text{ wenn }}j\neq k\right\}}$

to be viewed as.

If the function is monotonically increasing, it is called the content ${\displaystyle F\colon \mathbb {R} \rightarrow \mathbb {R} }$

${\displaystyle \mu _{F}\colon {\mathcal {J}}\rightarrow \mathbb {R} ,\mu _{F}(]a,b]):=F(b)-F(a),(a\leq b)}$

the related Stieltjesian content . It is σ-finite . ${\displaystyle F}$

Presentation of content

Is a finite content and is defined by ${\displaystyle \mu \colon {\mathcal {J}}\rightarrow \mathbb {R} }$ ${\displaystyle F:\mathbb {R} \rightarrow \mathbb {R} }$

${\displaystyle F(x):={\begin{cases}\mu (]0,x]),&{\mbox{wenn }}x\geq 0\\-\mu (]x,0]),&{\mbox{wenn }}x<0\end{cases}}}$,

so is a monotonically increasing function and it holds . With this, every finite content can be represented as Stieltjesian content. ${\displaystyle F}$ ${\displaystyle \mu =\mu _{F}}$ ${\displaystyle \mathbb {R} }$

Lebesgue-Stieltjes premise

One is often interested in whether a content is σ-additive , i.e.

${\displaystyle \mu _{F}\left(\bigcup _{i=1}^{\infty }I_{i}\right)=\sum _{i=1}^{\infty }\mu _{F}(I_{i})}$

applies if the pairs are different. Namely, σ-additive contents are premeasures and can be continued into dimensions. The Stieltjesian content is a pre-measure if and only if it is right-continuous . In this case it is called the Lebesgue-Stieltjes premise belonging to it. A special case arises for the Lebesgue pre-measure . If, on the other hand, one has chosen the half-ring of the intervals closed on the left as the system of quantities, then a premeasure is precisely when the left-hand side is continuous. This premeasure is also σ-finite. ${\displaystyle I_{i}}$ ${\displaystyle F}$ ${\displaystyle \mu _{F}}$ ${\displaystyle F}$ ${\displaystyle F(x)=x}$ ${\displaystyle \mu _{F}}$ ${\displaystyle F}$

Lebesgue-Stieltjes integral

With the help of the Stieltjes content, one can extend the Riemann integral to the Lebesgue-Stieltjes integral . For this purpose, the Carathéodory extension set is used to construct the Lebesgue-Stieltjes measure from the premise . The σ-finiteness of the measure provides the uniqueness of the continuation. The new integral term can finally be constructed from this measure. ${\displaystyle \textstyle \int _{a}^{b}g(x)\ \mathrm {d} F(x)}$

literature

• Wolfgang Walter : Analysis (=  basic knowledge of mathematics . Volume 4 ). Springer, Berlin a. a. 1990, ISBN 3-540-12781-X .
• Jürgen Elstrodt : Measure and integration theory . 4th, corrected edition. Springer, Berlin a. a. 2005, ISBN 3-540-21390-2 , Chapter II, § 2, p. 37 .