Locally finite measure

A locally finite measure in mathematics, more precisely in measure theory , is a mapping that assigns subsets of topological spaces to an abstracted volume. Local finiteness is an important property when investigating measures on topological spaces, because it guarantees the existence of a neighborhood with finite measure for every point.

definition

Consider a Hausdorff space and a σ-algebra , at least the Borel σ-algebra contains so . Then a measure is called ${\displaystyle (X,\tau )}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {B}}:=\sigma (\tau )}$ ${\displaystyle {\mathcal {A}}\supset {\mathcal {B}}}$

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

a locally finite measure if there is an open environment for each such that . ${\displaystyle x\in X}$ ${\displaystyle U_{x}}$ ${\displaystyle \mu (U_{x})<\infty }$

Examples

• Every finite measure is locally finite.
• The Lebesgue measure is locally finite, a possible open neighborhood of finite measure of would be for . ${\displaystyle x}$ ${\displaystyle (x-\epsilon ,x+\epsilon )}$ ${\displaystyle \epsilon >0}$

properties

If locally finite, then every compact set has finite measure. Because it is ${\displaystyle \mu }$

${\displaystyle K\subset \bigcup _{x\in K}U_{x}}$,

but due to the compactness there is a finite partial cover and thus ${\displaystyle (U_{x_{i}})_{i=1,\dots ,n}}$

${\displaystyle \mu (K)\leq \sum _{i=1}^{n}\mu (U_{x_{i}})<\infty }$.

If locally compact , then the converse also holds, i.e. that local is finite if and only if every compact set has finite measure. ${\displaystyle X}$ ${\displaystyle \mu }$

Related concepts

Borel dimensions

→ Main article : Borel measure

If a locally finite measure is defined on Borel's σ-algebra, it is also called a Borel measure. In the literature, however, there are numerous different concepts of Borel dimensions, some of which differ considerably. Therefore, a precise comparison with the corresponding definition is always necessary here.

Radon measures

→ Main article : Radon measure

A Radon measure is a locally finite measure on Borel's σ-algebra that is regular from within . From the inside it regularly means that for all true ${\displaystyle A\in {\mathcal {B}}}$

${\displaystyle \mu (A)=\sup\{\mu (K)\,|\,K\subset A,\,K{\text{ kompakt}}\}}$.

Like Borel measures, radon measures are not used uniformly in the literature; a comparison with the corresponding definitions is necessary.

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 .
• Achim Klenke : Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .