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$\sigma$-finite Borel and Radon measures (Topic)

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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Also defines:  $\sigma$-finite Borel measure, Radon measure
Keywords:  $\sigma$-finite Borel and Radon measures

Cross-references: topological, regular, Borel sets, sigma-algebra of Borel sets, locally compact Hausdorff space, field, Borel space

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Physics Classification: 02. (Mathematical methods in physics)
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