# 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

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

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}}$

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

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

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.

## 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).