Moderate measure

A moderate measure , also called moderate Borel measure , is a term from measure theory , a branch of mathematics that deals with generalized and abstracted concepts of length and volume and thus forms the basis for stochastics and integration theory .

Here, moderate measures are special measures on Hausdorff spaces that are Borel measures and for which there is a countable cover of the base space from open sets of finite measure. Moderate measures make it possible to specify more general criteria for the regularity of the measure , such as the finiteness of the Borel measure would allow.

Moderate dimensions were first introduced by Nicolas Bourbaki in 1969.

definition

A Hausdorff space is given and the associated Borel σ-algebra is given . A Borel measure ${\displaystyle (X,\tau )}$ ${\displaystyle {\mathcal {B}}=\sigma (\tau )}$

${\displaystyle \mu :{\mathcal {B}}\to [0,\infty ]}$

is called a moderate amount if there are open sets such that there is and for everyone ${\displaystyle (O_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mu (O_{n})<\infty }$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle X=\bigcup _{n\in \mathbb {N} }O_{n}}$ applies.

A measure as Borel measure is called when it finally locally is so if for every one around with are. ${\displaystyle x\in X}$ ${\displaystyle U_{x}}$ ${\displaystyle \mu (U_{x})<\infty }$

example

The Lebesgue measure on a moderate level, because it is locally finite. To do this, one chooses the environment for each point , then it is finite. A possible open coverage would be the quantities . ${\displaystyle \mathbb {R} }$ ${\displaystyle x}$ ${\displaystyle U_{x}=(x-\epsilon /2,x+\epsilon /2)}$ ${\displaystyle \lambda (U_{x})=\epsilon }$ ${\displaystyle O_{n}=(-n,n)}$

properties

• Every moderate measure is a σ-finite measure , because the requirement of an open coverage is a stronger requirement than the coverage with arbitrary quantities as it is required for the σ-finiteness. The reverse conclusion, i.e. from σ-finiteness to moderate measure, does not apply in general.
• Every externally regular σ-finite Borel measure is moderate. Because if there is a sequence of sets of finite measure that covers, then it follows from the regularity from outside that for each there is an open set with . Accordingly, they provide an open coverage with sets of finite measure as is required for a moderate measure. ${\displaystyle (M_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle M_{n}}$ ${\displaystyle U_{M_{n}}}$ ${\displaystyle \mu (M_{n})\leq \mu (U_{M_{n}})=\mu (M_{n})+\epsilon <\infty }$ ${\displaystyle U_{M_{n}}}$
• Every Borel measure in a σ-compact space is moderate. Because then there exist compact sets such that ${\displaystyle (K_{n})_{n\in \mathbb {N} }}$

${\displaystyle X=\bigcup _{n\in \mathbb {N} }K_{n}}$

is. According to the definition of compactness, there is a finite partial cover for the open cover of . Because of the local finiteness of the Borel measure, then for all and . So form the sets ${\displaystyle (U_{x}^{m})_{x\in K_{m}}}$ ${\displaystyle K_{m}}$ ${\displaystyle (U_{x_{i}}^{m})_{i\in I_{m}}}$ ${\displaystyle \mu (U_{x_{i}}^{m})<\infty }$ ${\displaystyle i\in I_{m}}$ ${\displaystyle m}$

${\displaystyle (O_{n})_{n\in \mathbb {N} }:=\left(\bigcup _{i\in I_{n}}U_{x_{i}}^{n}\right)_{n\in \mathbb {N} }}$

an open covering of sets of finite measure. ${\displaystyle X}$

• Every Borel measure on a Hausdorff space with a countable basis is moderate. This is shown by modifying a given basis so that it only contains sets of finite measure and then showing that it is still a basis. The countable number of base sets are then open by definition, each have only a finite measure and thus meet the requirements.
• Every Borel measure in a Lindelöf room is moderate. The local finiteness of provides an open covering of the space by sets of finite measure, the Lindelöf property now allows a countable partial covering to be selected from this. The two examples above are therefore a special case of this property. ${\displaystyle \mu }$

Moderate dimensions and regular dimensions

Moderate measures provide important regularity statements for Borel measures. In doing so, one exploits the fact that for an open finite cover the Borel measure is restricted to finite and thus many regularity properties of finite Borel measures are transferred to moderate Borel measures. ${\displaystyle (O_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mu }$ ${\displaystyle {\mathcal {B}}|_{O_{n}}}$

On Hausdorff rooms

For example, in Hausdorff spaces, if there is a moderate Borel measure and every open set with finite measure is regular from the inside, then that is also regular. ${\displaystyle \mu }$ ${\displaystyle \mu }$

With the above properties it immediately follows that the following conclusions apply to a Hausdorff space : ${\displaystyle X}$

1. If σ-compact, then every Borel measure in which every open set of finite measure is regular from inside is also regular. ${\displaystyle X}$
2. It follows directly from this that every radon measure on σ-compact is moderate and regular. Here, a radon measure denotes a Borel measure that is regular from the inside. ${\displaystyle X}$
3. If every open set is σ-compact, then every Borel measure is moderate and regular. Because every σ-compact set is regular from within.

On Polish rooms

According to Ulam's theorem , every Borel measure in a Polish area is regular and moderate.

Individual evidence

1. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 381.

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , pp. 318-322 , doi : 10.1007 / 978-3-540-89728-6 .