External dimension

External measure is a term from the mathematical branch of measure theory , which was introduced by Constantin Carathéodory in 1914 . External dimensions play an important role in the expansion of pre-dimensions to dimensions using the Carathéodory dimension expansion kit . External dimensions are generally not dimensions.

definition

An external measure is a set function of the power set of a set in the interval , which satisfies the following axioms : ${\displaystyle \nu }$ ${\displaystyle X}$ ${\displaystyle [0,\infty ]}$

• ${\displaystyle \nu (\emptyset )=0}$
• ${\displaystyle A\subseteq B\Rightarrow \nu (A)\leq \nu (B)}$    " Monotony "
• ${\displaystyle \nu \left(\bigcup _{i=1}^{\infty }A_{i}\right)\leq \sum _{i=1}^{\infty }\nu (A_{i})}$    " -Subadditivity " ${\displaystyle \sigma }$

The name external measure is based on the terms internal and external measure, which were used by Borel and Lebesgue . Carathéodory's theory uses no internal measure and simplifies the basic proofs considerably.

Metric outer dimension

A metric outer dimension is an outer dimension on a metric space with the additional property: ${\displaystyle (X,d)}$

• ${\displaystyle \nu (A\cup B)=\nu (A)+\nu (B)}$

for all non-empty separated sets and , d. H. . For example, when constructing the Lebesgue measure , a metric outer measure is used. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \inf\{d(a,b):a\in A,b\in B\}>0}$ ${\displaystyle \lambda }$ ${\displaystyle \lambda ^{*}}$

construction

External dimensions

Let be any set system with and a set function with . If one sets for all : ${\displaystyle S\subseteq {\mathcal {P}}(X)}$ ${\displaystyle \emptyset \in S}$ ${\displaystyle \mu \colon S\rightarrow [0,+\infty ]}$ ${\displaystyle \mu (\emptyset )=0}$ ${\displaystyle A\subseteq X}$

${\displaystyle \nu (A):={\begin{cases}\inf \ \left\{\left.\sum _{i=1}^{\infty }\mu (A_{i})\,\right|\,A_{i}\in S,\ A\subseteq \bigcup _{i=1}^{\infty }A_{i}\right\}\\+\infty ,{\text{ wenn }}A{\text{ in keiner abz}}\mathrm {\ddot {a}} {\text{hlbaren Vereinigung von Mengen aus }}S{\text{ enthalten ist}}.\end{cases}}}$

Then there is an external measure . If -subadditive, then applies to all . In this way, an outer dimension can be constructed in particular by means of a content or a pre-dimension on a half-ring or ring . Sometimes the above construction is therefore only defined for these special cases. ${\displaystyle \nu }$ ${\displaystyle {\mathcal {P}}(X)}$ ${\displaystyle \mu }$ ${\displaystyle \sigma }$ ${\displaystyle \mu (A)=\nu (A)}$ ${\displaystyle A\in S}$

If you choose the Lebesgue pre- measure as the pre -measure , you get the outer Lebesgue-Stieltjes measure , if you choose the Lebesgue-Stieltjes pre-measure , you get the outer Lebesgue-Stieltjes measure.

Metric outer dimensions

Let be any system of sets in the metric space with and a set function with . One defines ${\displaystyle S\subseteq {\mathcal {P}}(X)}$ ${\displaystyle (X,d)}$ ${\displaystyle \emptyset \in S}$ ${\displaystyle \mu \colon S\rightarrow [0,+\infty ]}$ ${\displaystyle \mu (\emptyset )=0}$

${\displaystyle \nu _{\delta }(A):={\begin{cases}\inf \ \left\{\left.\sum _{i=1}^{\infty }\mu (A_{i})\,\right|\,\operatorname {diam} (A_{i})\leq \delta ,\,A_{i}\in S,\ A\subseteq \bigcup _{i=1}^{\infty }A_{i}\right\}\\+\infty ,{\text{ wenn }}A{\text{ in keiner abzählbaren Vereinigung von Mengen aus }}S{\text{ mit Durchmesser }}\leq \delta {\text{ enthalten ist}},\end{cases}}}$

so is

${\displaystyle \nu (A):=\sup _{\delta >0}\nu _{\delta }(A)}$

a metric outer measure. Where is the diameter of the set . ${\displaystyle \operatorname {diam} (A)}$ ${\displaystyle A}$

In this way, for example, the outer Hausdorff dimension is defined, but the outer Lebesgue dimension can also be obtained in this way. To do this, one sets and and as a system of quantities, the half-ring of the half-open intervals. ${\displaystyle d(x,y)=|x-y|}$ ${\displaystyle \mu (A)=\operatorname {diam} (A)}$

Measurability according to Carathéodory

Let be an outer measure on the power of a set . A quantity is called measurable with respect to or briefly -measurable, if ${\displaystyle \nu \colon {\mathcal {P}}(X)\to [0,\infty ]}$ ${\displaystyle X}$ ${\displaystyle A\subseteq X}$ ${\displaystyle \nu }$ ${\displaystyle \nu }$

${\displaystyle \forall E\in {\mathcal {P}}(X):\nu (E)=\nu (E\cap A)+\nu (E\cap A^{c})}$.

This concept of measurability comes from Constantin Carathéodory . Equivalent is the definition that a quantity is measurable if and only if ${\displaystyle A\subseteq X}$ ${\displaystyle \nu }$

${\displaystyle \quad \nu (E)\geq \nu (E\cap A)+\nu (E\cap A^{c})}$applies to all . ${\displaystyle E\subseteq X}$

The two characterizations are equivalent, since the equals sign follows from the σ-subadditivity of the external measure.

Examples

• ${\displaystyle \emptyset ,X}$are measurable. ${\displaystyle \nu }$
• Complements of measurable quantities are measurable: Be measurable. Then it is also measurable. ${\displaystyle \nu }$ ${\displaystyle A\subseteq X}$ ${\displaystyle \nu }$ ${\displaystyle A^{c}}$ ${\displaystyle \nu }$
• Zero quantities with regard to the external dimension are measurable: Let with . Then -measurable. In the same way, -measurable, if applies. ${\displaystyle A\subseteq X}$ ${\displaystyle \nu (A)=0}$ ${\displaystyle A}$ ${\displaystyle \nu }$ ${\displaystyle A}$ ${\displaystyle \nu }$ ${\displaystyle \nu (A^{c})=0}$

Differentiation from other measurability terms

When a set can be measured, it is usually meant that this set is in a certain σ-algebra . This concept of measurability is mainly dependent on the measuring room in which one is located. Therefore, one also speaks in part of the measurability with regard to a measuring room .

In contrast to this, the concept of measurability used here is independent of a set system. It only depends on the external measure that is defined on the entire power set. Accordingly, according to Carathéodory, measurability is also called measurability in terms of an external dimension .

σ-algebra of ν-measurable sets

If there is an external measure, so is the quantity ${\displaystyle \nu }$

${\displaystyle {\mathcal {A}}_{\nu }=\{A\subset X|A{\text{ ist }}\nu {\text{-messbar}}\}}$

a σ-algebra and a complete measure . ${\displaystyle \nu |_{{\mathcal {A}}_{\nu }}}$

It can also show that if and the Borel σ-algebra contains when a metric outer measure on the metric space is. ${\displaystyle {\mathcal {A}}_{\nu }}$ ${\displaystyle {\mathcal {B}}(X)}$ ${\displaystyle \nu }$ ${\displaystyle (X,d)}$

See also

Measure , measurable quantities

literature

• Jürgen Elstrodt : Measure and integration theory. 4th, corrected edition. Springer, Berlin et al. 2005, ISBN 3-540-21390-2 , Chapter II § 4.1.
• Heinz Bauer : Measure and integration theory. 2nd, revised edition. de Gruyter, Berlin et al. 1992, ISBN 3-11-013626-0 , § 5.