PlanetPhysics/Sigma Finite Borel and Radon Measures

Introduction

Let us recall the following data related to [[../BorelSpace/|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 [[../CosmologicalConstant/|field]] of real numbers

$ℝ$ generated by the open intervals of $ℝ$;

1. [[../BorelSpace/|Borel space]]
2. Consider a [[../LocallyCompactHausdorffSpaces/|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 $ℬ(X)$ 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

${\left\{K_n\right\}}_n$ of compact subsets $K_n$ of $X$ such that:

$X=\bigcup _{n=1}^{\mathrm{\infty }}{K}_n.$

Definition: Borel Space

Let $(X;ℬ(X))$ be a Borel space (with the $\sigma$-algebra $ℬ(X)$ 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 $A_n\in ℬ(X)$ for all $n$, such that $\bigcup _{n=1}^{\mathrm{\infty }}{A}_n=X,$ and also $\mu (A_n)<\mathrm{\infty }$ 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).