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
is called a Borel measure if for each there is an open environment of with .
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
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.
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.
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 .
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
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 .
Radon measures
Borel measures are called radon measures if they are regular from the inside , so it applies that
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.
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
- ↑ Elstrodt: Measure and Integration Theory. 2009, p. 313.
- ^ VV Sazonov: Borel Measure . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- ^ 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 .
- ↑ Eric W. Weisstein : Borel Measure . In: MathWorld (English).