Lebesgue Stieltjes measure

The Lebesgue-Stieltjes measure is a term from measure theory , a sub-area of ​​mathematics. It contains the Lebesgue measure as a special case and is used to construct the Lebesgue-Stieltjes integral .

definition

Given a monotonically increasing , right continuous function and the measuring room , with the Borel σ-algebra called. Then the unambiguously determined dimension is called on this measuring room ${\displaystyle F\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle \lambda _{F}}$

${\displaystyle \lambda _{F}((a,b]):=F(b)-F(a)\quad {\text{für}}\quad a<b}$

Lebesgue Stieltjes measure.

Examples

• The best-known example of a Lebesgue-Stieltjes measure is the Lebesgue measure , from which the Lebesgue integral is constructed. Here is . ${\displaystyle \lambda ((a,b])=b-a}$ ${\displaystyle F(x)=x}$
• For and with for and for , the Lebesgue-Stieltjes measure is the Dirac measure . ${\displaystyle a\in \mathbb {R} }$ ${\displaystyle F\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle F(x)=0}$ ${\displaystyle x<a}$ ${\displaystyle F(x)=1}$ ${\displaystyle x\geq a}$ ${\displaystyle \lambda _{F}}$ ${\displaystyle \delta _{a}}$
• If a nonnegative, continuous function with an antiderivative , then the measure is with density . ${\displaystyle f\colon \mathbb {R} \to [0,\infty )}$ ${\displaystyle F}$ ${\displaystyle \lambda _{F}}$ ${\displaystyle f}$
• If in addition and , then is a probability measure and is the distribution function . ${\displaystyle \lim _{x\to -\infty }F(x)=0}$ ${\displaystyle \lim _{x\to \infty }F(x)=1}$ ${\displaystyle \lambda _{F}}$ ${\displaystyle F}$
• If both of the above cases are met, it is a probability measure with density. These measures play an important role in stochastics .

construction

Let the half-ring and a growing, right-sided continuous function be given . Then ${\displaystyle {\mathcal {J}}:=\{(a,b]:a,b\in \mathbb {R} ,a<b\}}$ ${\displaystyle F}$

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

a σ-finite premise , the so-called Lebesgue-Stieltjessches premise . Then, with the extension set from Carathéodory, a clear continuation of this pre-measure can be constructed into a measure. For this purpose, an external measure , the so-called external Lebesgue-Stieltjes measure, is defined and this is restricted to the σ-algebra generated by . This σ-algebra is then exactly Borel's σ-algebra and it is . ${\displaystyle \nu _{F}}$ ${\displaystyle {\mathcal {J}}}$ ${\displaystyle {\mathcal {B}}(\mathbb {R} )=\sigma ({\mathcal {J}})}$ ${\displaystyle \lambda _{F}=\nu _{F}|\sigma ({\mathcal {J}})}$

completion

The dimension space constructed above is generally not a complete dimension space . Since the Lebesgue-Stieltjes external measure is also a metric external measure , the σ-algebra of the measurable sets contains the Borel σ-algebra with regard to the external measure . Hence the measure space is the completion of . ${\displaystyle {\mathcal {A}}_{\nu _{F}}}$ ${\displaystyle (\mathbb {R} ,{\mathcal {A}}_{\nu _{F}},\nu _{F}|_{{\mathcal {A}}_{\nu _{F}}})}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ),\lambda _{F})}$

literature

• Jürgen Elstrodt: Measure and integration theory. 6th edition. Springer, Berlin / Heidelberg / New York 2009, ISBN 978-3-540-89727-9 .
• Achim Klenke: Probability Theory. 2nd Edition. Springer, Berlin / Heidelberg 2008, ISBN 978-3-540-76317-8 .