Definition:Radon Measure

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: $\S 5.3$