Carrier (measurement theory)

The carrier of a measure is a term from measure theory , a branch of mathematics that deals with generalized volume concepts. Similar to the carrier of a function in analysis, the compactness of the carrier guarantees certain properties such as the integrability of continuous functions.

definition

Let a Hausdorff space and a Radon measure (in the sense of an internally regular , locally finite measure ) be given , the Borel σ-algebra . ${\displaystyle (X,\tau )}$ ${\displaystyle \mu }$ ${\displaystyle {\mathcal {B}}:=\sigma (\tau )}$

If the (possibly uncountable) family of open -null sets, then is ${\displaystyle ({\mathcal {O}}_{i})_{i\in I}}$ ${\displaystyle \mu }$

${\displaystyle O:=\bigcup _{i\in I}O_{i}}$

the largest open zero set in terms of set-theoretical inclusion . Your complement is called the carrier of , so ${\displaystyle \mu }$ ${\displaystyle \mu }$

${\displaystyle \operatorname {supp} (\mu )=X\setminus O}$.

Alternatively, there is also the notation . ${\displaystyle \operatorname {Tr} (\mu )}$

comment

The fact that there really is a null set can be seen as follows: If and compact, there is a finite coverage of by the definition of compactness . So is due to the monotony of measure . But since the inside is regular, it follows . ${\displaystyle O}$ ${\displaystyle K\subset O}$ ${\displaystyle O_{i_{k}}}$ ${\displaystyle K}$ ${\displaystyle \mu (K)\leq \sum _{k=1}^{n}\mu (O_{i_{k}})=0}$ ${\displaystyle \mu }$ ${\displaystyle \mu (O)=0}$

properties

• If the carrier of a Radon measure is compact, then all continuous functions can be integrated, i.e. is ${\displaystyle \mu }$ ${\displaystyle C(X)\subset {\mathcal {L}}^{1}(\mu )}$
• Conversely, on a σ-compact , locally compact Hausdorff space, in which applies to a radon measure , the carrier of this radon measure is always compact. ${\displaystyle C(X)\subset {\mathcal {L}}^{1}(\mu )}$ ${\displaystyle \mu }$

Web links

• MI Voitsekhovskii: Support of a measure . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .