PlanetPhysics/Sigma Finite Borel and Radon Measures

Introduction
Let us recall the following data related to Borel space and measure theory: $$\mathbb{R}$$ generated by the open intervals of $$\mathbb{R}$$; $$\left\{K_n \right\}_n$$ of compact subsets $$K_n$$ of $$X$$ such that:
 * 1) sigma-algebra, or $\sigma$-algebra;
 * 2) the Borel algebra  which is defined as the smallest $$\sigma$$-algebra on the field of real numbers
 * 1) Borel space
 * 2) 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 $$X$$;
 * 3) When the Borel measure $$\mu$$ is both inner and outer regular on all Borel sets, it is called a Borel measure;
 * 4) Recall that a topological space $$X$$ is $$\sigma$$-compact if there exists a sequence
 * $$X = \bigcup_{n=1}^\infty K_n .$$

Definition: Borel Space
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 $$\bigcup_{n=1}^\infty A_n = X,$$ and also $$\mu(A_n) < \infty $$ for all $$n$$, (ref. ).

Definition: Radon Measure
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).