Definition:Radon Measure
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Definition
Let $\struct {X, \tau}$ be a Hausdorff space.
Let $\map \BB X$ denote the Borel $\sigma$-algebra generated by $\tau$.
Let $\MM$ be a $\sigma$-algebra over $X$ such that $\map \BB X \subseteq \MM$.
Let $\mu : \MM \to \overline \R$ be a measure on $\MM$, where $\overline \R$ denotes the set of extended real numbers.
Then $\mu$ is called a Radon measure, if and only if:
- $(1): \quad \map \mu K < \infty$ for all compact sets $K \subseteq X$
- $(2): \quad \map \mu B = \sup \leftset {\map \mu K : K \subseteq B, K}$ is compact for all $\rightset {B \in \MM}$.
Also see
- Results about Radon measures can be found here.
Source of Name
This entry was named for Johann Karl August Radon.
Sources
- 2001: Christian Berg and Tage Gutmann Madsen: Mål- og integralteori: $\S 5.3$