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[parent] $\sigma$--finite Borel measure and related Borel concepts (Topic)

Preliminary Data

Let us recall the following data related to Borel space and measure theory:
  1. sigma-algebra, or $ \sigma$-algebra;
  2. the Borel algebra which is defined as the smallest $ \sigma$-algebra on the field of real numbers $ \mathbb{R}$ generated by the open intervals of $ \mathbb{R}$;
  3. Borel space
  4. Consider a locally compact Hausdorff space $ X$; a Borel measure is then defined as any measure $ \mu$ on the sigma-algebra of Borel sets, that is, the Borel $ sigma$-algebra $ \mathcal{B}(X)$ defined on a locally compact Hausdorff space $ X$;
  5. When the Borel measure $ \mu$ is both inner and outer regular on all Borel sets, it is called a regular Borel measure.
Definition 0.1   Let $ (X; \mathcal{B}(X))$ be a Borel space (with the $ \sigma$-algebra $ \mathcal{B}(X)$ of Borel sets of a topological space $ X$), and let $ \mu$ be a measure on the space $ X$. Then, such a measure is called a $ \sigma$-finite (Borel) measure if there exists a sequence $ \left\{A_n \right\}_n$ with $ A_n \in \mathcal{B}(X)$ for all $ n$, such that
$\displaystyle \bigcup_{n=1}^\infty A_n = X,$
and also $ \mu(A_n) < \infty $ for all $ n$, (ref. [1]).

Remark: If $ \mu$ is an inner regular and locally finite measure, then $ \mu$ is said to be a Radon measure.

Note: Recall that a topological space $ X$ is $ \sigma$-compact if there exists a sequence $ \left\{K_n \right\}_n$ of compact subsets $ K_n$ of $ X$ such that

$\displaystyle X = \bigcup_{n=1}^\infty K_n .$
Then, any Borel measure on $ X$ which is finite on such compact subsets is also (Borel) $ \sigma$-finite in the above defined sense (Definition 0.1).

Bibliography

1
M.R. Buneci. 2006., Groupoid C*-Algebras., Surveys in Mathematics and its Applications, Volume 1: 71-98.
2
J.D. Pryce (1973). Basic methods of functional analysis., Hutchinson University Library. Hutchinson, p. 212-217.
3
Alan J. Weir (1974). General integration and measure. Cambridge University Press, pp. 150-184.
4
Boris Hasselblatt, A. B. Katok, Eds. (2002). Handbook of Dynamical Systems., vol. 1A, p.678. North-Holland. on line



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See Also: Borel space, Borel set, outer regular, Borel measure, $\sigma$-algebra, groupoid, groupoids, groupoid C*-convolution algebras, Borel G-space

Other names:  $sigma$-finite Borel measure
Also defines:  regular Borel measure, Borel algebra, Borel sigma-algebra, Radon measure, sigma-finite Borel measure
Keywords:  sigma-finite Borel measure, Borel space

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Cross-references: finite, compact subsets, locally finite, inner regular, sequence, topological space, Borel sets, sigma-algebra, Borel measure, locally compact Hausdorff space, open intervals, generated by, real numbers, field, theory, measure, Borel space
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This is version 20 of $\sigma$--finite Borel measure and related Borel concepts, born on 2008-09-20, modified 2008-11-11.
Object id is 11053, canonical name is SigmaFiniteBorelMeasureAndRelatedBorelConcepts.
Accessed 471 times total.

Classification:
AMS MSC: 28A10 (Measure and integration :: Classical measure theory :: Real- or complex-valued set functions)
 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)
 54H05 (General topology :: Connections with other structures, applications :: Descriptive set theory )
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