$\sigma$-finite Borel and Radon measures

Introduction

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;
  6. 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 :

    $\displaystyle X = \bigcup_{n=1}^\infty K_n .$

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]).

Definition 0.2   If $ \mu$ is an inner regular and locally finite measure, then $ \mu$ is said to be a Radon measure.

Note 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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