Borel measure

A Borel measure is a term from measure theory , a branch of mathematics that deals with generalized volume concepts. Borel measures are clearly distinguished by the fact that each point can be enveloped in a set of finite measure and that they are defined on a special σ-algebra. Borel measures are important basic concepts when examining measures in topological spaces. They are named after Émile Borel .

Caution is advised when using Borel measures, as these are not uniformly defined in the literature, especially in the Anglo-Saxon language area.

definition

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

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

is called a Borel measure if for each there is an open environment of with . ${\displaystyle x\in X}$ ${\displaystyle U_{x}}$ ${\displaystyle x}$ ${\displaystyle \mu (U_{x})<\infty }$

Thus Borel measures are locally finite measures on Borel's σ-algebra. A special case of this is the Lebesgue-Borel measure .

Other meanings

The term is not used consistently in the specialist literature. Sometimes too

• outer dimensions with respect to which all Borel amounts Carathéodory-measurable are
• arbitrary measures on the Borel σ-algebra of a topological space

• the measure on the Borel σ-algebra , which assigns the measure to each interval ${\displaystyle \mathbb {R} }$ ${\displaystyle [a,b]}$ ${\displaystyle b-a}$

referred to as Borel measure. The measure in the third case is usually called the Borel-Lebesgue measure .

Unless otherwise stated, this article discusses the properties of Borel measures in the sense given in the definition above.

properties

For a locally compact Hausdorff space , the local finitude is equivalent to the fact that every compact set has finite measure. ${\displaystyle X}$

Because , because of the local compactness to an environment, there is a compact and an open environment of with . Local finitude now follows from the monotony of measure; it is then and is open as required. ${\displaystyle x\in X}$ ${\displaystyle U_{x}}$ ${\displaystyle K_{x}}$ ${\displaystyle O_{x}}$ ${\displaystyle x}$ ${\displaystyle O_{x}\subset K_{x}\subset U_{x}}$ ${\displaystyle \mu (O_{x})\leq \mu (K_{x})<\infty }$ ${\displaystyle O_{x}}$

Conversely, it follows from local finiteness that every compact set has finite measure: Let be an open neighborhood of with . Then there is an open cover of . From the definition of compactness it follows that a finite partial coverage exists; so is . ${\displaystyle K}$ ${\displaystyle x\in K}$ ${\displaystyle O_{x}}$ ${\displaystyle x}$ ${\displaystyle \mu (O_{x})<\infty }$ ${\displaystyle (O_{x})_{x\in K}}$ ${\displaystyle K}$ ${\displaystyle (O_{x_{i}})_{i\in I},\;|I|<\infty }$ ${\displaystyle \mu (K)\leq \sum _{i\in I}\mu (O_{x})<\infty }$

This property is also used to define Borel measures on locally compact Hausdorff spaces, but in the general case does not agree with local finiteness.

Related concepts

Moderate dimensions

A Borel measure is called a moderate measure if there is a sequence of open sets such that ${\displaystyle (O_{n})_{n\in \mathbb {N} }}$

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

is and applies to everyone . Moderate measures are of particular interest because more general criteria apply to them, among which a Borel measure is a regular measure . ${\displaystyle \mu (O_{n})<\infty }$ ${\displaystyle n\in \mathbb {N} }$

Radon measures

Borel measures are called radon measures if they are regular from the inside , so it applies that

${\displaystyle \mu (A)=\sup\{\mu (K)\mid K\subset A,\ K\ {\textrm {kompakt}}\}}$

for everyone . Like Borel measures, the term "radon measure" is not used uniformly in the literature and should therefore always be compared with the exact definition in the given context. ${\displaystyle A\in {\mathcal {B}}}$

Regular Borel dimensions

A Borel measure is called a regular Borel measure if it is also a regular measure . Thus, every externally regular radon measure is a regular Borel measure. However, since there are separate regularity terms for each use of the term "Borel measure", caution is required here too and a comparison with the definitions in the respective context 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 .

Individual evidence

1. ↑ Elstrodt: Measure and Integration Theory. 2009, p. 313.
2. ^ VV Sazonov: Borel Measure . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
3. ^ Lawrence C. Evans , Ronald F. Gariepy: Measure Theory and Fine Properties of Functions . CRC-Press, Boca Raton FL et al. 1992, ISBN 0-8493-7157-0 .
4. ↑ Eric W. Weisstein : Borel Measure . In: MathWorld (English).