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, :


 * $(1): \quad \map \mu K < \infty$ 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 $\rightset {B \in \MM}$.